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Градиентный бустинг. Реализация с нуля на Python и разбор особенностей его модификаций (XGBoost, CatBoost, LightGBM)

На сегодняшний день градиентный бустинг (gradient boosting machine) является одним из основных production-решений при работе с табличными, неоднородными данными, поскольку обладает высокой производительностью и точностью, а если быть точнее, то его модификации, речь о которых пойдёт чуть позже.

В данной статье представлена не только реализация градиентного бустинга GBM с нуля на Python, но а также довольно подробно описаны ключевые особенности его наиболее популярных модификаций.

Ноутбук с данными алгоритмами можно загрузить на Kaggle (eng) и GitHub (rus).

Содержание

Gradient Boosting Machine

Gradient Boosting Machine

Как и в случае с Adaboost, градиентный бустинг добавляет базовые модели в ансамбль последовательно, однако вместо обучения моделей с учётом весов на основе ошибок предшественников, в данном случае модели обучаются на остаточных ошибках (residual errors), допущенных предыдущими моделями.

Принцип работы градиентного бустинга для регрессии

Алгоритм строится следующим образом:

1) первоначальному прогнозу присваивается среднее значение y_train для всех образцов;
2) рассчитываются остатки модели на основе антиградиента функции потерь;
3) регрессионное дерево обучается на X_train и остатках, далее делается прогноз на X_train;
4) полученный прогноз добавляется к первоначальному и шаги 2-4 повторяются для каждого дерева;
5) после обучения всех моделей снова создаётся первоначальный прогноз из шага 1;
6) далее делаются прогнозы для X_test на обученных деревьях и добавляются к первоначальному;
7) полученная сумма и будет конечным прогнозом.

Формулы для расчётов

Функция потерь: $L(y_i, F(x_i)) = \frac{1}{2} (y_i - F(x_i))^2$

Остатки: $r_{ik} = -\left[\frac{\partial L(y_i, F(x_i))}{\partial F(x_i)}\right]_ {F(x) = F_{m-1}(x)} = y_{i} - F_{m-1}(x_i)$

Принцип работы градиентного бустинга для классификации

В случае классификации дела обстоят немного сложнее: к каждому уникальному классу необходимо применить one-hot encoding и перевести в вероятности с помощью softmax-функции, а к прогнозам деревьев добавить коэффициент $\gamma$ , который регулирует степень вклада каждого нового дерева в общую модель для снижения переобучения. Также качество прогнозов градиентного бустинга можно существенно улучшить, применив концепцию K-class LogitBoost: для каждого дерева рассчитываются веса, а только потом на их основе остатки.

Алгоритм строится следующим образом:

1) для y_train применяется one-hot encoding и первоначальному прогнозу приcваиваются значения 0 для каждого класса;
2) прогнозы преобразуются в вероятности с помощью softmax-функции;
3) рассчитываются остатки модели на основе антиградиента функции потерь и вероятностей (а также веса в случае LogitBoost);
4) регрессионное дерево обучается на X_train и остатках (а в случае LogitBoost ещё присваиваются веса), далее делается прогноз на X_train;
5) для каждого листа в дереве рассчитываются коэффициенты $\gamma$ на основе остатков, взятых по позициям наблюдений, попавших в определённый в листовой узел;
6) полученные прогнозы для каждого класса и сумма коэффициентов $\gamma$ добавляются к первоначальным;
7) шаги 2-6 повторяются для каждого дерева в каждом классе;
8) после обучения всех моделей создаётся первоначальный прогноз из шага 1;
9) далее делаются прогнозы для X_test на обученных деревьях по каждому классу и добавляются к первоначальным;
10) классы с максимальной суммой и будут конечным прогнозом.

Формулы для расчётов

Функция потерь:

$L(\{y_k, F_k(x)\}_{1}^{K}) = -\sum\limits_{k=1}^{K} y_k \ log \ p_k(x)$

Вероятность каждого класса:

$p_k(x) = softmax(F_k(x)) = \frac{e^{F_k(x)}}{\sum\limits_{l=1}^{K} e^{F_l(x)}}$

Остатки:

$r_{ik} = -\left[\frac{\partial L(\{y_{il}, F_l(x_i)\}_{l=1}^{K})}{\partial F_k(x_i)}\right]_ {\{F_l(x) = F_{l,m-1}(x)\}_{1}^{K}} = y_{ik} - p_{k,m-1}(x_i)$

Коэффициент $\gamma$ :

$\gamma_{jkm} = \frac{K-1}{K} \frac{\sum_{x_i \in R_{jkm}} r_{ik}} {\sum_{x_i \in R_{jkm}} |r_{ik}| (1 - |r_{ik}|)}$

Веса для K-class LogitBoost:

Остатки для K-class LogitBoost:

$r_{ik} = \frac{K-1}{K} \frac{y_{ik} - p_k(x_i)}{w_k(x_i)}$

Импорт необходимых библиотек

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import load_diabetes
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score, mean_absolute_percentage_error
from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import GradientBoostingClassifier, GradientBoostingRegressor
from sklearn.ensemble import HistGradientBoostingClassifier, HistGradientBoostingRegressor
from mlxtend.plotting import plot_decision_regions

Реализация на Python с нуля

class GBMClassifier:
    def __init__(self, logitboost=False, learning_rate=0.1, n_estimators=100,
                 max_depth=3, random_state=0):
        self.logitboost = logitboost
        self.learning_rate = learning_rate
        self.n_estimators = n_estimators
        self.max_depth = max_depth
        self.random_state = random_state

    def _softmax(self, predictions):
        exp = np.exp(predictions)

        return exp / np.sum(exp, axis=1, keepdims=True)

    def _compute_gammas(self, residuals, leaf_indexes, eps=1e-4):
        gammas = []

        for j in np.unique(leaf_indexes):
            x_i = np.where(leaf_indexes == j)
            numerator = np.sum(residuals[x_i])
            norm_residuals_xi = np.linalg.norm(residuals[x_i]) + eps
            denominator = np.sum(norm_residuals_xi * (1 - norm_residuals_xi))
            gamma = (self.K - 1) / self.K * numerator / denominator
            gammas.append(gamma)

        return gammas

    def fit(self, X, y):
        self.K = len(np.unique(y))
        self.trees = {k: [] for k in range(self.K)}
        one_hot_y = pd.get_dummies(y).to_numpy()   # one-hot encoding
        predictions = np.zeros(one_hot_y.shape)

        for _ in range(self.n_estimators):
            probabilities = self._softmax(predictions)

            for k in range(self.K):
                if self.logitboost:   # based on K-class LogitBoost
                    numerator = (one_hot_y.T[k] - probabilities.T[k])
                    denominator = probabilities.T[k] * (1 - probabilities.T[k])
                    residuals = (self.K - 1) / self.K * numerator / denominator
                    weights = denominator
                else:
                    residuals = one_hot_y.T[k] - probabilities.T[k]
                    weights = None

                tree = DecisionTreeRegressor(criterion='friedman_mse', max_depth=self.max_depth,
                                             random_state=self.random_state)
                tree.fit(X, residuals, sample_weight=weights)
                self.trees[k].append(tree)

                leaf_indexes = tree.apply(X)
                gammas = [] if self.logitboost else self._compute_gammas(residuals, leaf_indexes)
                predictions.T[k] += self.learning_rate * tree.predict(X) + np.sum(gammas)

    def predict(self, samples):
        predictions = np.zeros((len(samples), self.K))

        for i in range(self.n_estimators):
            for k in range(self.K):
                predictions.T[k] += self.learning_rate * self.trees[k][i].predict(samples)

        return np.argmax(predictions, axis=1)

class GBMRegressor:
    def __init__(self, learning_rate=0.1, n_estimators=100, max_depth=3, random_state=0):
        self.learning_rate = learning_rate
        self.n_estimators = n_estimators
        self.max_depth = max_depth
        self.random_state = random_state
        self.trees = []

    def fit(self, X, y):
        self.initial_leaf = y.mean()
        predictions = np.zeros(len(y)) + self.initial_leaf

        for _ in range(self.n_estimators):
            residuals = y - predictions
            tree = DecisionTreeRegressor(criterion='friedman_mse', max_depth=self.max_depth,
                                         random_state=self.random_state)
            tree.fit(X, residuals)
            predictions += self.learning_rate * tree.predict(X)
            self.trees.append(tree)

    def predict(self, samples):
        predictions = np.zeros(len(samples)) + self.initial_leaf

        for i in range(self.n_estimators):
            predictions += self.learning_rate * self.trees[i].predict(samples)

        return predictions

Код для отрисовки решающих границ

def decision_boundary_plot(X, y, X_train, y_train, clf, feature_indexes, title=None):
    feature1_name, feature2_name = X.columns[feature_indexes]
    X_feature_columns = X.values[:, feature_indexes]
    X_train_feature_columns = X_train.values[:, feature_indexes]
    clf.fit(X_train_feature_columns, y_train.values)

    plot_decision_regions(X=X_feature_columns, y=y.values, clf=clf)
    plt.xlabel(feature1_name)
    plt.ylabel(feature2_name)
    plt.title(title)

Загрузка датасетов

Для обучения моделей будет использован Glass Classification датасет, где необходимо верно определить тип стекла по его признакам. В случае регрессии используется Diabetes датасет из scikit-learn.

Данные ниже будут использоваться в примерах для всех модификаций градиентного бустинга.

df_path = "/content/drive/MyDrive/glass.csv"
glass_df = pd.read_csv(df_path)
X1, y1 = glass_df.iloc[:, :-1], glass_df.iloc[:, -1]
y1 = pd.Series(LabelEncoder().fit_transform(y1))
X1_train, X1_test, y1_train, y1_test = train_test_split(X1, y1, random_state=0)
print(glass_df)


          RI     Na    Mg    Al     Si     K    Ca    Ba   Fe  Type
0    1.52101  13.64  4.49  1.10  71.78  0.06  8.75  0.00  0.0     1
1    1.51761  13.89  3.60  1.36  72.73  0.48  7.83  0.00  0.0     1
2    1.51618  13.53  3.55  1.54  72.99  0.39  7.78  0.00  0.0     1
3    1.51766  13.21  3.69  1.29  72.61  0.57  8.22  0.00  0.0     1
4    1.51742  13.27  3.62  1.24  73.08  0.55  8.07  0.00  0.0     1
..       ...    ...   ...   ...    ...   ...   ...   ...  ...   ...
209  1.51623  14.14  0.00  2.88  72.61  0.08  9.18  1.06  0.0     7
210  1.51685  14.92  0.00  1.99  73.06  0.00  8.40  1.59  0.0     7
211  1.52065  14.36  0.00  2.02  73.42  0.00  8.44  1.64  0.0     7
212  1.51651  14.38  0.00  1.94  73.61  0.00  8.48  1.57  0.0     7
213  1.51711  14.23  0.00  2.08  73.36  0.00  8.62  1.67  0.0     7

[214 rows x 10 columns]

X2, y2 = load_diabetes(return_X_y=True, as_frame=True)
X2_train, X2_test, y2_train, y2_test = train_test_split(X2, y2, random_state=0)
print(X2, y2, sep='\n')


          age       sex       bmi        bp        s1        s2        s3  \
0    0.038076  0.050680  0.061696  0.021872 -0.044223 -0.034821 -0.043401   
1   -0.001882 -0.044642 -0.051474 -0.026328 -0.008449 -0.019163  0.074412   
2    0.085299  0.050680  0.044451 -0.005670 -0.045599 -0.034194 -0.032356   
3   -0.089063 -0.044642 -0.011595 -0.036656  0.012191  0.024991 -0.036038   
4    0.005383 -0.044642 -0.036385  0.021872  0.003935  0.015596  0.008142   
..        ...       ...       ...       ...       ...       ...       ...   
437  0.041708  0.050680  0.019662  0.059744 -0.005697 -0.002566 -0.028674   
438 -0.005515  0.050680 -0.015906 -0.067642  0.049341  0.079165 -0.028674   
439  0.041708  0.050680 -0.015906  0.017293 -0.037344 -0.013840 -0.024993   
440 -0.045472 -0.044642  0.039062  0.001215  0.016318  0.015283 -0.028674   
441 -0.045472 -0.044642 -0.073030 -0.081413  0.083740  0.027809  0.173816   

           s4        s5        s6  
0   -0.002592  0.019907 -0.017646  
1   -0.039493 -0.068332 -0.092204  
2   -0.002592  0.002861 -0.025930  
3    0.034309  0.022688 -0.009362  
4   -0.002592 -0.031988 -0.046641  
..        ...       ...       ...  
437 -0.002592  0.031193  0.007207  
438  0.034309 -0.018114  0.044485  
439 -0.011080 -0.046883  0.015491  
440  0.026560  0.044529 -0.025930  
441 -0.039493 -0.004222  0.003064  

[442 rows x 10 columns]
0      151.0
1       75.0
2      141.0
3      206.0
4      135.0
       ...  
437    178.0
438    104.0
439    132.0
440    220.0
441     57.0
Name: target, Length: 442, dtype: float64

Обучение моделей и оценка полученных результатов

В случае классификации ручная реализация немного отличается в плане точности от scikit-learn — это связано с тем, что в последней применяется более усложнённый подход для прогнозирования и расчёта остатков, но суть остаётся прежней, и в большинстве случаев работает примерно также как и ручная реализация, особенно в сравнении с LogitBoost модификацией.

В случае регрессии, ошибка в ручной реализации даже немного ниже, чем в scikit-learn — различия незначительны.

Также стоит отметить, что в scikit-learn есть особая версия градиентного бустинга — HistGradientBoosting на основе LightGBM, предназначенная для работы с большими наборами данных, которая отрабатывает гораздо быстрее и чаще лучше, чем классический градиентный бустинг.

Полученные результаты приведены ниже.

GBMClassifier

gbc = GBMClassifier(random_state=0)
gbc.fit(X1_train, y1_train)
gbc_pred_res = gbc.predict(X1_test)
gbc_accuracy = accuracy_score(gbc_pred_res, y1_test)
print(f'gbc accuracy: {gbc_accuracy}')
print(gbc_pred_res)


gbc accuracy: 0.6111111111111112
[5 0 1 4 1 1 0 1 1 1 1 0 1 1 1 5 0 1 1 1 5 0 5 5 1 0 5 0 1 0 0 4 1 0 0 0 0
 0 0 5 1 4 1 0 0 1 0 1 0 1 0 3 5 0]

GBMClassifier с концепцией LogitBoost

logit_bc = GBMClassifier(logitboost=True, random_state=0)
logit_bc.fit(X1_train, y1_train)
logit_bc_pred_res = logit_bc.predict(X1_test)
logit_bc_accuracy = accuracy_score(logit_bc_pred_res, y1_test)
print(f'logit_gbc accuracy: {logit_bc_accuracy}')
print(logit_bc_pred_res)


logit_gbc accuracy: 0.6666666666666666
[5 0 1 4 1 1 0 1 1 1 1 0 0 1 1 5 2 1 1 0 3 0 5 1 2 0 5 0 0 1 0 0 1 0 0 0 0
 2 0 5 1 4 1 0 1 1 0 1 0 1 0 1 5 0]

GradientBoostingClassifier (scikit-learn)

sk_gbc = GradientBoostingClassifier(random_state=0)
sk_gbc.fit(X1_train, y1_train)
sk_gbc_pred_res = sk_gbc.predict(X1_test)
sk_gbc_accuracy = accuracy_score(sk_gbc_pred_res, y1_test)
print(f'sk_gbc accuracy: {sk_gbc_accuracy}')
print(sk_gbc_pred_res)

feature_indexes = [1, 3]
title1 = 'GradientBoostingClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_gbc, feature_indexes, title1)


sk_gbc accuracy: 0.6666666666666666
[5 0 1 4 1 1 0 1 1 1 1 0 1 1 1 5 2 1 2 0 1 0 5 5 0 0 5 0 0 1 0 0 1 0 1 0 0
 0 0 5 1 1 1 0 1 1 0 1 0 1 0 1 5 0]

HistGradientBoostingClassifier (scikit-learn)

sk_hist_gbc = HistGradientBoostingClassifier(random_state=0)
sk_hist_gbc.fit(X1_train, y1_train)
sk_hist_gbc_pred_res = sk_hist_gbc.predict(X1_test)
sk_hist_gbc_accuracy = accuracy_score(sk_hist_gbc_pred_res, y1_test)
print(f'sk_hist_gbc accuracy: {sk_hist_gbc_accuracy}')
print(sk_hist_gbc_pred_res)

feature_indexes = [1, 3]
title2 = 'HistGradientBoostingClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_hist_gbc, feature_indexes, title2)


sk_hist_gbc accuracy: 0.7407407407407407
[5 0 1 4 1 1 0 1 1 1 0 1 0 1 1 5 2 1 0 2 3 0 5 5 0 0 5 0 1 1 0 0 1 0 0 0 0
 2 0 5 1 4 1 0 0 1 0 1 0 1 0 1 5 0]

GBMRegressor

gbr = GBMRegressor(random_state=0)
gbr.fit(X2_train, y2_train)
gbr_pred_res = gbr.predict(X2_test)
mape = mean_absolute_percentage_error(gbr_pred_res, y2_test)
print(f'gbr mape: {mape}')
print(gbr_pred_res)


gbr mape: 0.3422881238725283
[255.26058697 236.87383554 189.22369747 111.4291292  195.31958776
 249.2293251   97.46956406 205.91110801 133.47952304 240.88679883
 187.19018483 171.09963906 132.63933991  86.28964353 286.56077589
 105.78347251 161.94686043  72.58069452 105.92470096 236.52718059
 178.31146418 127.93036968 164.16563703 147.68681371 229.72747212
 194.16656023 142.44300193  68.79177464 237.21638657 163.36730299
 228.0338107   85.82843049 122.42781509 153.99332065 133.44848601
 159.00754731 151.98981608 147.1238119   88.56372077 200.84785209
 111.32129613 157.60108633 137.94395538 173.39402287 176.29284757
  62.85019316 109.646804   124.14982054  83.45697583 266.40618239
 141.2307366   42.05033124 163.57898214 149.92213387 230.37326368
 226.98842404 176.95426024 128.73302071  94.7641018  180.13853236
 242.31821245 147.52903035 130.45931832  97.32617479 246.70709949
 155.26338705  88.36785447 243.39330218 215.93042351  57.89243255
  87.76236479 138.21222825 101.23449416 140.80521012 128.63575744
 180.15062966 126.14000544 220.28212417 272.05642136 203.4384734
 135.96206295 180.87259854  55.62066017 241.84218009 137.19320157
  85.57441147 154.16763601 198.15465153 107.34659467 158.72672412
 102.17554103 114.4602227   70.44016611 165.66282053 108.02991765
  87.88855231 218.47152891 250.1208086  131.93276827 164.8880371
 174.16393905 119.21512394 150.9405975   93.44225978 234.7444088
 139.46581273 248.28728947 276.8291188  121.36081625  77.12248695
 252.1021757 ]

GradientBoostingRegressor (scikit-learn)

sk_gbr = GradientBoostingRegressor(random_state=0)
sk_gbr.fit(X2_train, y2_train)
sk_gbr_pred_res = sk_gbr.predict(X2_test)
sk_mape = mean_absolute_percentage_error(sk_gbr_pred_res, y2_test)
print(f'sk_gbr mape: {sk_mape}')
print(sk_gbr_pred_res)


sk_gbr mape: 0.34315075601365186
[255.26058697 236.87383554 189.22369747 111.4291292  195.31958776
 249.2293251   97.46956406 205.91110801 133.47952304 240.88679883
 187.19018483 171.09963906 132.63933991  86.28964353 286.56077589
 105.78347251 161.94686043  72.58069452 105.92470096 236.52718059
 182.08417504 127.93036968 164.16563703 147.68681371 229.72747212
 194.16656023 142.44300193  68.79177464 237.21638657 163.36730299
 223.14955275  85.82843049 122.42781509 153.99332065 133.44848601
 159.00754731 151.98981608 147.1238119   88.56372077 200.84785209
 117.71059889 157.60108633 137.94395538 173.39402287 176.29284757
  62.85019316 109.646804   124.14982054  83.45697583 266.40618239
 141.2307366   42.05033124 163.57898214 149.92213387 230.37326368
 226.98842404 176.95426024 128.73302071  94.7641018  180.13853236
 242.31821245 147.52903035 131.71442058  97.32617479 246.70709949
 155.26338705  88.36785447 243.39330218 215.93042351  54.04270734
  87.76236479 138.21222825 101.23449416 140.80521012 128.63575744
 180.15062966 126.14000544 220.28212417 272.05642136 203.4384734
 135.96206295 180.87259854  55.62066017 241.84218009 137.19320157
  85.57441147 154.16763601 198.15465153 107.34659467 158.72672412
 102.17554103 114.4602227   70.44016611 165.66282053 108.02991765
  87.88855231 218.47152891 247.93843617 131.93276827 164.8880371
 174.16393905 119.21512394 150.9405975   93.44225978 234.7444088
 139.46581273 248.28728947 286.90898175 121.36081625  77.12248695
 252.1021757 ]

HistGradientBoostingRegressor (scikit-learn)

sk_hist_gbr = HistGradientBoostingRegressor(random_state=0)
sk_hist_gbr.fit(X2_train, y2_train)
sk_hist_gbr_pred_res = sk_hist_gbr.predict(X2_test)
sk_hist_mape = mean_absolute_percentage_error(sk_hist_gbr_pred_res, y2_test)
print(f'sk_hist_gbr mape: {sk_hist_mape}')
print(sk_hist_gbr_pred_res)


sk_hist_gbr mape: 0.3625770694795175
[264.59007545 243.59190524 180.62442773 108.53165744 195.17695513
 258.1710931  112.67894454 239.18006052 144.50027011 244.02461776
 180.19087697 171.2628973  142.6966707   94.26236013 301.44493762
  90.25622203 154.82566511  75.43291346  96.36987316 230.14565012
 196.50928346 134.41863815 167.28719774 118.51218378 209.75431111
 176.46248808 113.40188129  68.47515467 248.57638999 116.15751993
 195.54397416  59.94733366 153.03012884 165.84819847 135.95402508
 133.54279177 190.26058998 153.08963938  86.65214731 185.07312005
 110.00197459 142.99434304 111.32566034 205.63519803 163.76035901
  68.26003406 118.15074567 101.01137587  57.54356002 247.29839792
 140.32714196  41.87878548 149.89720974 189.08931404 251.48016218
 175.76119949 160.38118455 110.76716142 107.00114228 154.75409577
 256.3194536  148.8935956  119.32752592  78.65118163 252.18315433
 122.833807    92.10033912 219.56310647 199.35553751  87.27114699
  78.83529151 114.14943246  86.07433503 110.44443265 126.1152782
 193.58960355 125.06171193 207.62012728 297.41734601 206.57210109
 121.92170504 194.37322797  69.66978916 260.97214455  99.77944002
 104.34927662 131.51094535 200.0857846  114.70076625 144.92225838
 107.08233694  84.70304021  65.20782844 156.77203859  93.87016441
 117.10361355 213.79169982 258.68468943 129.45477593 140.06443861
 176.72311312 107.95798342 172.45179867 115.75603211 240.83612061
 102.2339954  249.6969811  293.80624063 110.47825419  73.86099524
 238.42985813]

Преимущества и недостатки градиентного бустинга (GBM)

Преимущества:

высокая точность;
хорошо работает с неоднородными данными;
лежит в основе других модификаций градиентного бустинга.

Недостатки:

низкая скорость обучения при работе с большими датасетами;
отсутствие возможности работы с категориальными признаками;
возможность переобучения из-за плохой предобработки данных.

Стоит отметить, что вышеперечисленные недостатки были устранены частично либо полностью в модификациях градиентного бустинга, про которые сейчас и пойдёт речь.

Дополнительные источники

Статья «Greedy Function Approximation: A Gradient Boosting Machine», Jerome H. Friedman.

Документация:

Лекции:

Пошаговое построение Gradient Boosting:

Особенности XGBoost

XGBoost (Extreme Gradient Boosting) является, пожалуй, самой популярной модификацией градиентного бустинга на сегодняшний день, с помощью которой было выиграно множество соревнований на Kaggle. Рассмотрим причины по которым так произошло.

В отличие от GBM в XGboost изначальный прогноз регрессионного дерева устанавливается f_0 (x) = 0.5 и вместо пней с глубиной 3 берутся более глубокие деревья с листьями в диапазоне от 8 до 32 как в случае регрессии, так и классификации, а к функции потерь добавляется регуляризация (L1 или L2 на выбор), которая штрафует дерево с высокой суммой нормы значений в листьях. Данная сумма представлена в виде целевой функции, расчёт значений которой аппроксимируется разложением в ряд Тейлора 2-го порядка.

Формулы для расчётов

Целевая функция:

$Obj^{\ t} = \sum\limits_{i=1}^{N} L(y_i, \hat{y_i}^{(t-1)} + f_t(x_i)) + \Omega(f_t)$

L2-регуляризация:

$\Omega(f_t) = \gamma \ T + \frac{1}{2} \lambda||w||^2$

Аппроксимация целевой функции разложением в ряд Тейлора 2-го порядка:

$Obj^{\ t} \approx \sum\limits_{i=1}^{N} [L(y_i, \hat{y_i}^{(t-1)}) + g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i) ] + \gamma \ T + \frac{1}{2} \lambda \sum\limits_{j=1}^{T} w_j^2 = \\ = \sum\limits_{j=1}^{T} [(\sum\limits_{i \in I_j}g_i) w_j + \frac{1}{2} (\sum\limits_{i \in I_j}h_i + \lambda) w_j^2 ] + \gamma \ T$

Где:

градиент первого порядка: ;
градиент второго порядка (гессиан): ;
параметры регуляризации: $\gamma, \lambda$ .

Приближение второго порядка удобно с вычислительной точки зрения, поскольку большинство членов одинаковы на каждой отдельной итерации и большая часть выражения может быть вычислена один раз и повторно использована как константа для всех остальных разбиений, и в таком случае единственное, что остаётся вычислить — это f_t(x_i) и $\Omega(f_t)$ .

Оптимальное значение веса j-го листового узла в дереве рассчитывается следующим образом:

$w^*_j = -\frac{\sum_{i \in I_j} g_i}{\sum_{i \in I_j} h_i + \lambda}$

Мера загрязнённости представляет из себя качество структуры дерева q и рассчитывается следующим образом:

$Obj^{\ t} = -\frac{1}{2} \sum\limits_{j=1}^{T} \frac{(\sum_{i \in I_j} g_i)^2}{\sum_{i \in I_j} h_i + \lambda} + \gamma \ T$

Чем меньше это значение, тем лучше структура дерева.

Поскольку перечисление всевозможных древовидных структур и выбор наилучшей весьма осложнён, мы можем оптимизировать данную задачу, добавляя по одному разбиению узла дерева за раз. В частности, мы итеративно добавляем ветви к дереву, используя жадный алгоритм, а информационный прирост будет выглядеть следующим образом:

$gain = \frac{1}{2} \left[\frac{(\sum_{i \in I_L} g_i)^2}{\sum_{i \in I_L} h_i + \lambda} + \frac{(\sum_{i \in I_R} g_i)^2}{\sum_{i \in I_R} h_i + \lambda} - \frac{(\sum_{i \in I} g_i)^2}{\sum_{i \in I} h_i + \lambda} \right] - \gamma$

Информационный прирост будет показывать изменение чистоты узла после его разбиения. Его максимальное значение будет соответствовать наилучшему пороговому значению (критерию разбиения) узла в дереве.

Стоит также упомянуть, что в XGBoost поддерживается выбор линейных моделей в качестве базовых, но, в целом, они работают хуже деревьев.

Дополнительные оптимизации в XGBoost

Weighted quantile sketch

В случае работы с большими наборами данных в XGBoost используется приближённый жадный алгоритм. Это означает, что вместо тестирования всех возможных пороговых значений мы тестируем только квантили. Используя распараллеливание для разделения набора данных и создания ветвей в каждом дереве независимо друг от друга чтобы несколько ядер могли работать с ними одновременно, алгоритм взвешенных квантилей (weighted quantile sketch) объединяет данные в приближённую гистограмму, разделённую на взвешенные квантили, которые помещают наблюдения с прогнозами низкой достоверности в квантили с меньшим количеством наблюдений.

Cache-aware access & out-of-core computing

Для более быстрого вычисления прогнозов градиенты и гессианы хранятся в кэш-памяти (cache-aware access), а в случае с датасетами большого размера, когда недостаточно кэш и оперативной памяти, задействуется память из жёсткого диска. Учитывая, что чтение данных на жёстком диске выполняется гораздо медленнее чем в кэш-памяти, для ускоренной обработки производится их компрессия. Данные хранятся в единицах памяти, называемых блоками, информация в которых хранится в формате сжатых столбцов, отсортированных по соответствующим значениям параллельно с использованием всех доступных ядер ЦП, а распаковка каждого блока осуществляется в независимом потоке при загрузке в память. Данная техника называется "вычисление вне ядра" (out-of-core computing). При наличии нескольких жёстких дисков, применяется параллельное сегментирование блоков для каждого жёсткого диска (sharding).

При использовании сжатых данных необходимо учитывать, что процесс сжатия и распаковки занимает дополнительное время, поэтому выбор алгоритма сжатия (с потерями и без) и его параметров может повлиять на производительность в целом, однако это всё равно значительно быстрее чтения без сжатия.

Sparsity-aware split finding

В XGboost также поддерживается работа с пропусками. Суть заключается в следующем: из исходного датасета убираются наблюдения с пропусками и представляются как отдельный датасет. После построения дерева для наблюдений без пропусков рассчитываются их остатки, далее — остатки для наблюдений с пропусками, которые присоединяются к остаткам дерева как для левого, так и для правого узла. В конце рассчитывается информационный прирост (gain) для каждого узла в дереве: выбирается порог для разбиения с максимальным приростом. Данная техника называется "поиск разбиения с учётом разряженности" (sparsity-aware split finding).

Обучение на GPU

Для ускорения процесса обучения в XGBoost реализована поддержка GPU с помощью параметров device="cuda" и tree_method="gpu_hist". Также стоит отметить, что в данном алгоритме поддерживается полностью распределённое обучение на GPU с использованием Dask, Spark и PySpark.

Пакетная реализация XGBoost и оценка полученных результатов

Благодаря вышеуказанным оптимизациям и особенностям XGBoost работает лучше классической реализации, что видно на примере классификации, однако на примере регрессии в данном случае результат оказался хуже. Если хорошо подготовить данные и поэкспериментировать с настройкой гиперпараметров, XGBoost будет работать гораздо лучше как в случае регрессии, так и классификации.

XGBClassifier

from xgboost import XGBClassifier, XGBRegressor

xgbc = XGBClassifier(seed=0)
xgbc.fit(X1_train, y1_train)
xgbc_pred_res = xgbc.predict(X1_test)
xgbc_accuracy = accuracy_score(xgbc_pred_res, y1_test)
print(f'xgbc accuracy: {xgbc_accuracy}')
print(xgbc_pred_res)

feature_indexes = [1, 3]
title2 = 'XGBClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, xgbc, feature_indexes, title2)


xgbc accuracy: 0.6851851851851852
[5 0 1 4 1 1 0 1 1 1 3 0 1 1 1 5 2 1 1 2 3 0 5 5 0 0 5 0 0 1 0 0 1 0 0 0 0
 0 0 5 1 4 1 0 1 1 0 1 0 1 0 1 5 0]

XGBRegressor

xgbr = XGBRegressor(seed=0)
xgbr.fit(X2_train, y2_train)
xgbr_pred_res = xgbr.predict(X2_test)
xgbr_mape = mean_absolute_percentage_error(xgbr_pred_res, y2_test)
print(f'xgbr mape: {xgbr_mape}')
print(xgbr_pred_res)


xgbr mape: 0.3781388070785457
[281.3304   241.21165  212.13124  136.89372  144.54208  261.7871
 125.257454 236.16089  199.79964  250.18103  188.50882  126.724464
 161.2112    73.98386  302.50272   97.35103  160.85774   67.10394
  96.0406   284.72873  181.48717  117.764626 182.91393  127.76686
 229.30229  173.58592  170.03932   76.73085  240.7698   117.70699
 224.36049   63.628925 124.27327  142.07358  173.09418  139.11557
 163.61208   92.8561    95.91142  217.65016  112.03522  122.800545
 110.06953  181.34201  159.39062   61.97588  114.534805 145.20259
  84.905815 284.77567  195.29976   45.550606  92.54941  180.4689
 279.12097  241.1343   177.77785  130.43887  159.53189  158.2313
 253.25339  181.86093  120.71679   90.604706 274.5059   140.21674
  92.80314  211.71936  233.20622  110.791824  67.89235  133.62137
 123.05923  102.70884  167.2115   159.32262  131.7305   243.41493
 254.97531  210.0554   130.00981  222.21094   52.864147 251.23187
 118.602684  87.89112  121.147095 218.15724   90.9274    91.69777
 101.95428   85.79966   79.59963  155.879    118.25745   99.5975
 231.70824  189.6652   121.97507  147.51166  152.10051  102.239395
 177.74615  107.55845  271.80927  121.41044  232.97308  292.89096
 129.39447   60.501278 259.46295 ]

Преимущества и недостатки XGBoost

Преимущества:

относительно высокая скорость обучения и точность прогнозов;
возможность работы с пропусками;
гибкость в конфигурации из-за большого количества гиперпараметров.

Недостатки:

сложность в конфигурации из-за большого количества гиперпараметров;
на данный момент отсутствует полноценная (присутствует экспериментальная) возможность работы с категориальными признаками, что может сильно снизить производительность при кодировании больших объёмов данных.

Дополнительные источники

Статья «XGBoost: A Scalable Tree Boosting System», Tianqi Chen, Carlos Guestrin.

Документация.

Лекции:

один;
два.

Пошаговое построение XGBoost:

Особенности CatBoost

CatBoost (Categorical Boosting) — более эффективная реализация градиентного бустинга от компании Яндекс, особенностью которой является возможность работы с категориальными признаками без их предварительной обработки. Разберёмся подробнее как это происходит.

Ordered target (CatBoost) encoding

При кодировании категориальных признаков к бустингам применяется один из нескольких классических подходов:

target encoding;
one-hot encoding (хорошо работает лишь в том случае, когда в признаках содержится небольшое количество значений);
feature hashing в несколько корзин (исходные данные преобразуются в уникальную последовательность символов (хэш-значения), которые разбиваются на группы (корзины) и обрабатываются параллельно с последующим применением target или one-hot encoding);
mean target encoding (рассчитывается новое значение для текущего категориального признака как среднее значение целевой переменной по датасету);
leave-one-out (рассчитывается среднее значение целевой переменной по всем объектам кроме текущего);
leave-bucket-out (рассчитывается среднее значение целевой переменной на одной части данных и применяется для оставшихся других);

Однако все вышеперечисленные способы так или иначе приводят к утечке данных и переобучению, и данную проблему можно частично решить с применением mean target encoding на отложенной выборке, что работает заметно лучше, но такой способ не идеален так как остаётся меньше данных для обучения и вычисления статистик по категориальным признакам.

В CatBoost применяется модифицированный подход к mean target encoding, называемый Ordered target (CatBoost) encoding, который состоит из 3-х нижеперечисленных этапов.

Случайная перестановка

Образцы в наборе данных перемешиваются случайным образом, что позволяет создать более сбалансированный набор данных и снизить переобучение.

Статистики «по прошлому»

После случайной перестановки рассчитывается новое значение для текущего категориального признака следующим образом:

$CTR = \frac{TargetSum + Prior}{TotalCount + 1}$

Prior — константа, определяемая как среднее арифметическое целевых значений.

TargetSum — сумма целевых значений для конкретного категориального признака до текущего момента.

TotalCount — количество всех наблюдений для конкретного категориального признака до текущего момента.

Данная формула может быть применима как для регрессии, так и для мультиклассификации.

Пошаговый пример с нуля для мультиклассификации

!pip install category_encoders

from category_encoders.cat_boost import CatBoostEncoder

train = pd.DataFrame({"color": ["red", "blue", "black", "black", "blue", "blue", "red", "blue"]})
target = pd.Series([2, 1, 2, 1, 0, 3, 0, 1])

cbe_encoder = CatBoostEncoder()
train_cbe = cbe_encoder.fit_transform(train, target)
prior = target.mean()

print(f'prior: {prior}', '', sep='\n')
print(train_cbe)


prior: 1.25

    color
0  1.2500
1  1.2500
2  1.2500
3  1.6250
4  1.1250
5  0.7500
6  1.6250
7  1.3125

В случае c числовыми признаками создаётся набор меток с помощью квантования и производится их бинаризация на k классов: все значения до границы (метки) относятся к классу 0, а после к классу 1. Далее процедура схожа со случаем выше.

В CatBoost поддерживаются следующие виды квантования:

мода;
медианная сетка;
равномерная сетка + квантили (размер квантования уменьшается вдвое и разбиения (пороговые значения) объединяются по одному из двух способов: медиана или равномерная сетка);
MaxLogSum — максимизируется значение следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} log \ w_i$

MinEntropy — минимизируется значение следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} w_i \ log \ w_i$

GreedyLogSum — максимизируется аппроксимация жадным способом следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} log \ w_i$

где n — число различных объектов в сегменте, w (weight) — число повторений объекта в сегменте.

Комбинация категориальных признаков

В CatBoost используется комбинация категориальных признаков (например, красный диван, синий диван, синий стул и так далее), что позволяет модели учитывать неочевидные взаимосвязи между ними и улучшить качество итоговых прогнозов в целом. Для избегания экспоненциального роста комбинаций категориальных признаков, их создание происходит не перед началом обучения, а в момент построения каждого дерева жадным алгоритмом отдельно.

Структура решающих деревьев и упорядоченный бустинг

В качестве базовых моделей в CatBoost используются симметричные односторонние деревья, имеющие одинаковую глубину и ширину. Создатели алгоритма считают, что такой подход позволяет ускорить обучение и создавать деревья, которые менее подвержены переобучению.

Для нахождения наилучшего критерия разбиения в узле используется косинусное сходство — его максимальное значение будет соответствовать лучшему критерию:

$Cosine \ Similarity = \frac{ \sum\limits_{i=1}^{n} A_i B_i}{\sqrt{\sum\limits_{i=1}^{n} A_i^2} \sqrt{\sum\limits_{i=1}^{n} B_i^2}}$

где A — вектор с остаточными ошибками, B — вектор с прогнозами в листовых узлах.

Проблема классического бустинга заключается в том, что прогнозы базовых моделей могут иметь высокое смещение, что в конечном счёте приводит к переобучению модели в целом — это обусловлено тем, что обучение и оценка деревьев производится на одинаковом наборе данных (X_train). Данная проблема решается с помощью метода линейной аппроксимации упорядоченного бустинга (Ordered Boosting), суть которого заключается в следующем: данные перемешиваются случайным образом и после чего текущее дерево обучается на m-1 образцах с учётом остатков предыдущего, что очень похоже на принцип в ordered target encoding, то есть изначально берётся один образец из набора данных и модель учится на всех образцах до него, то есть на нулях, далее берётся два образца и модель учится на одном, после чего рассчитываются остатки, которые будут использованы для следующей модели и так далее. Такой подход позволяет произвести обучение дерева со сложностью O(log(m)) вместо O(m^2) .

Структура симметричного одностороннего дерева

Дополнительные возможности CatBoost

Работа с пропусками

Поддерживаются следующие режимы обработки пропусков:

"Forbidden" — пропуски интерпретируются как ошибки.
"Min" — пропуски обрабатываются как минимальное значение (меньшее, чем все остальные значения) для объекта.
"Max" — пропуски обрабатываются как максимальное значение (превышающее все остальные значения) для объекта.

Режим обработки по умолчанию — Min. Также при выборе деревьев учитывается разбиение, отделяющее пропущенные значения от всех остальных.

Snapshots

Данный параметр (save_snapshot=True) позволяет создавать резервные копии промежуточных результатов. Если происходит неожиданное прерывание обучения (например, случайное выключение компьютера), то оно может быть продолжено из сохранённого состояния и в таком случае не нужно повторять завершённые итерации построения деревьев.

Обучение на GPU

В CatBoost возможно обучение не только на нескольких GPU, но и сразу на нескольких серверах с GPU, что фактически позволяет запустить распределённое обучение и использовать датасеты любого размера. Данная процедура позволяет значительно ускорить работу алгоритма: чем больше датасет, тем больше будет прирост в производительности в сравнении с обучением на CPU. Для того чтобы этим воспользоваться достаточно указать task_type="GPU".

Пакетная реализация CatBoost и оценка полученных результатов

Как можно заметить из примера классификации, не смотря на большое количество улучшений, CatBoost далеко не всегда показывает высокие результаты в сравнении с другими реализациями градиентного бустинга, как это заявляют его разработчики, особенно заметна разница в случае с несбалансированными наборами данных. Однако, стоит отметить, что в случае хорошей подготовки данных, при прочих равных, в ряде случаев CatBoost будет отличаться от других модификаций немного более высокой точностью.

В случае регрессии, когда нет явно выраженной проблемы как в случае выше, CatBoost справляется довольно хорошо.

Полученные результаты приведены ниже.

!pip install catboost

CatBoostClassifier

from catboost import CatBoostClassifier, CatBoostRegressor

cb_clf = CatBoostClassifier(random_seed=0, logging_level='Silent')
cb_clf.fit(X1_train, y1_train)
cb_clf_pred_res = cb_clf.predict(X1_test).squeeze()   # remove all single-dimensional entries
cb_clf_accuracy = accuracy_score(cb_clf_pred_res, y1_test)
print(f'cb_clf accuracy: {cb_clf_accuracy}')
print(cb_clf_pred_res)

feature_indexes = [1, 3]
title3 = 'CatBoostClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, cb_clf, feature_indexes, title3)


cb_clf accuracy: 0.6851851851851852
[5 0 1 4 1 1 0 1 1 1 0 0 1 1 1 5 2 1 2 2 1 0 5 1 0 0 5 0 1 1 0 0 1 0 0 0 0
 0 0 5 1 5 1 0 1 1 0 1 0 1 0 1 5 0]

CatBoostRegressor

cb_reg = CatBoostRegressor(random_seed=0, logging_level='Silent')
cb_reg.fit(X2_train, y2_train)
cb_reg_pred_res = cb_reg.predict(X2_test)
cb_reg_mape = mean_absolute_percentage_error(cb_reg_pred_res, y2_test)
print(f'cb_reg mape: {cb_reg_mape}')
print(cb_reg_pred_res)


cb_reg mape: 0.3260573594870776
[268.21698587 226.61112865 152.56164473 105.39534082 185.56084388
 231.13386917 112.94031582 246.11217136 121.83631454 238.66069067
 193.92810529 171.53207065 120.95427566  98.87431921 301.0159263
  97.989809   170.68360243  70.03044356 104.9534818  235.1624283
 174.7895105  131.39173163 178.28683674 128.13631321 230.60611298
 186.37090017 131.52230822  61.453026   231.95028492 148.67901568
 186.2317954   80.45908918 137.68617262 171.24445717 128.77648933
 178.31364898 179.17236177 149.82094832 102.25676058 209.48231728
 116.01443961 169.06621701 110.9047463  210.24240477 158.28102362
  72.4532758  135.86931928 121.14865532  76.8103627  278.66222581
 127.85077736  55.25704176 144.51194393 170.06468442 271.51905928
 187.61879537 159.56680649 118.65520621 121.92004997 183.56316238
 248.87847858 137.77180529 117.98743897  82.53109543 255.59388802
 151.23062005  94.00606836 257.15481144 227.21705762  96.77322241
  78.34198817 118.09701594  91.2069645   91.70192792 125.65538331
 155.68222799 129.63411563 230.46496307 292.38275237 213.68057983
 131.76653094 217.93105223  50.37188156 256.16334997  94.4107044
  90.95547399 139.58356594 187.26517    127.97009716 171.85831366
  89.65051176  95.93129442  60.14437246 178.27466698 101.36657382
 106.15753799 206.11327567 217.74079838 130.83283251 138.63144499
 154.18156801 123.38315127 197.65233246 125.95249034 239.25024631
 139.69849648 245.02582928 288.17892899 122.76437963  70.023686
 233.36946028]

Преимущества и недостатки CatBoost

Преимущества:

высокая точность и скорость обучения в большинстве случаев;
возможность работы с категориальными признаками;
большое количество гиперпараметров для настройки алгоритма;
применяется для многих задач.

Недостатки:

нестабильная работа с несбалансированными данными;
сложность в конфигурации из-за большого количества гиперпараметров;
плохая документация (личное мнение).

Дополнительные источники

Статья «CatBoost: unbiased boosting with categorical features», Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, Andrey Gulin.

Документация.

Лекция.

Пошаговое построение CatBoost (в ordered target encoding есть ошибки):

один;
два.

Особенности LightGBM

LightGBM (Light Gradient Boosting Machine) — реализация градиентного бустинга от компании Microsoft, способная обрабатывать большие наборы данных и при этом обеспечивать высокую точность. Рассмотрим более подробно ключевые особенности, которые на это повлияли.

Gradient-based One-Side Sampling (GOSS)

В отличие от классического градиентного бустинга, где регрессионные деревья обучаются на всём множестве X_train, в LightGBM деревья обучаются на подмножестве данных, полученном с помощью одностороннего отбора на основе градиента (GOSS), суть которого заключается в сортировке и выборе образцов top N с наибольшим градиентом — это означает, что именно этим данным следует уделить особое внимание, поскольку на них дерево будет ошибаться больше всего. Помимо этого, случайным образом производится отбор части образцов с низким градиентом с последующим присвоением им весов $\frac{1-a}{b}$ , чтобы избежать изменений в распределении данных. В данном случае — доля образцов top N в процентах с наибольшим градиентом, — доля образцов в процентах с низким градиентом.

Проще говоря, использование GOSS позволяет значительно ускорить обучение каждого последующего дерева в ансамбле путём уменьшения обучающего набора данных без особой потери в качестве прогнозов в большинстве случаев.

Histogram-based tree node splitting

При поиске наилучшего критерия разбиения узла в LightGBM также можно использовать гистограмму значений признаков: перебираются все пороговые значения гистограммы вместо всех значений признаков, что позволяет существенно ускорить поиск наилучшего критерия. Например, для признака "рост" можно использовать вопросы "рост" ⩽ 160, "рост" ⩽ 170 вместо "рост" ⩽ 163, "рост" ⩽ 167 и так далее. Как можно было заметить, гистограммы применяются в том или ином виде во всех трёх модификациях градиентного бустинга.

Exclusive Feature Bundling (EFB)

Предположим, что тем или иным образом в датасете имеются закодированные категориальные признаки, большинство из которых являются взаимоисключающими (те, которые одновременно не принимают ненулевые значения). При большом числе таких признаков обучение модели может занять слишком много времени и для решения данной проблемы в LightGBM применяется понижение размерности пространства признаков путём объединения взаимоисключающих признаков (EFB).

Стоит отметить, что в действительности строго взаимоисключающие признаки в данных встречаются крайне редко. Поэтому вводится такое понятие как доля конфликтов между признаками — отношение наблюдений от общего числа, где два признака принимают одновременно ненулевые значения.

Допустим, имеется датасет с признаками F0−F4 . В первую очередь стоит установить допустимую долю конфликтов, например, 25%. Тогда в случае из 8 образцов допустимое число конфликтов будет составлять 2. Далее строится матрица конфликтов между всеми признаками и производится подсчёт общей суммы конфликтов для каждого признака, после чего происходит их сортировка в порядке убывания.

На основе матрицы конфликтов строится взвешенный граф, у которого в полученном порядке отсекаются все рёбра с числом конфликтов, превышающим допустимое значение.

Работа Exclusive Feature Bundling с нуля на простом примере

Первой связкой будет F_0−F_1 . Не смотря на то что F_1−F_2 и F_1−F_3 не конфликтуют, их объединение невозможно из-за конфликтов F_0−F_2 и F_0−F_3 . Следующей связкой будет F_2−F_4 . Не смотря на то, что F_4 и F_3 не конфликтуют, их объединение невозможно из-за конфликта F_2−F_3 . Последним остаётся признак F_3 .

Новая связка будет состоять из ненулевых значений первого признака и ненулевых значений второго плюс смещение, равное максимальному значению первого признака: смещение каждого следующего признака будет равно сумме максимальных значений предыдущих. В случаях, когда наблюдается конфликт, берётся значение последнего признака со смещением.

Подробные расчёты приведены ниже.

Нетрудно заметить, что алгоритм EFB можно оптимизировать, игнорируя вычисления для всех нулевых значений в признаках. Также данную оптимизацию можно применить к histogram-based tree node splitting, что в целом снизит сложность построения гистограммы с O(data) до O(non-zero data).

Leaf-wise tree growth

Вместо построения деревьев слой за слоем, в LightGBM деревья строятся в глубину: рост продолжается в листовом узле с наибольшим информационным приростом, в результате чего дерево получается глубоким и несимметричным. Чаще всего такой подход превосходит симметричные деревья в плане точности.

Обучение на GPU

При работе с большими датасетами в LightGBM поддерживается обучение не только на GPU, но и распределённое обучение сразу на нескольких машинах.

На данный момент существуют следующие подходы:

Data parallel (образцы датасета разбиваются на локальные гистограммы, каждая из которых обрабатывается отдельно);
Feature parallel (признаки датасета разбиваются на локальные гистограммы, каждая из которых обрабатывается отдельно);
Voting parallel (на каждом процессе локально выбирается K лучших признаков для разбиения, после чего выбирается наилучший из них глобально).

Пакетная реализация LightGBM и оценка полученных результатов

Как можно заметить, в случае классификации LightGBM показал результат из коробки значительно лучше, чем предыдущие модификации бустинга. Также стоит отметить, что из всех трёх рассмотренных модификацией градиентного бустинга LightGBM является самой быстрой: чем больше датасет, тем сильнее будет заметна разница в плане скорости.

Полученные результаты приведены ниже.

LGBMClassifier

from lightgbm import LGBMClassifier, LGBMRegressor

lgb_clf = LGBMClassifier(random_state=0, verbose=-1)
lgb_clf.fit(X1_train, y1_train)
lgb_clf_pred_res = lgb_clf.predict(X1_test)
lgb_clf_accuracy = accuracy_score(lgb_clf_pred_res, y1_test)
print(f'lgb_clf accuracy: {lgb_clf_accuracy}')
print(lgb_clf_pred_res)

feature_indexes = [1, 3]
title4 = 'LGBMClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, lgb_clf, feature_indexes, title4)


lgb_clf accuracy: 0.7407407407407407
[5 0 1 4 1 1 0 1 1 1 0 0 0 1 1 5 2 1 2 2 3 1 5 5 0 0 5 0 1 1 0 0 1 0 0 0 0
 0 0 5 1 4 1 0 0 1 0 1 0 1 0 1 5 0]

LGBMRegressor

lgb_reg = LGBMRegressor(random_state=0, verbose=-1)
lgb_reg.fit(X2_train, y2_train)
lgb_reg_pred_res = lgb_reg.predict(X2_test)
lgb_reg_mape = mean_absolute_percentage_error(lgb_reg_pred_res, y2_test)
print(f'lgb_reg_mape: {lgb_reg_mape}')
print(lgb_reg_pred_res)


lgb_reg_mape: 0.349719771087187
[261.46864967 223.24674049 179.60231224 104.76984808 197.67529058
 264.26116883 111.11355079 243.72059654 144.18045853 239.08131542
 199.90841705 170.37417433 151.67933516  89.30779664 297.93561211
  96.58862229 162.53285879  70.06113305 108.36436227 234.16276627
 207.7552967  136.83913826 172.38724529 110.26981866 228.1574526
 163.56817258 108.94516511  73.93303203 233.31869769 116.00042252
 195.39846082  72.54403358 164.38943926 194.42418913 130.42154381
 183.04435501 173.56352142 147.00774306  89.07365731 175.25759644
 108.72749406 144.48000593  99.60713604 196.86889512 160.47835096
  60.17798032 133.4909559  115.46543242  64.48286456 251.20520649
 135.52608509  45.16396653 161.70420785 187.28406745 274.21158331
 179.30355071 150.1563969  111.1275583  100.5067047  133.7868607
 269.98464504 159.03980859 118.01578638  75.41625416 249.0569831
 133.99026562 111.43954756 221.81954307 206.02676942  81.56098377
  72.5920367  101.31664062  93.31261471 111.24487763 149.40048547
 177.37205369 130.08118649 212.0573     294.138509   232.24330644
 109.5457663  181.3487432   63.20386576 261.47508264  88.43007908
  99.59568927 127.97978786 211.77672798 121.44351586 138.27282283
 121.01455597  91.26965383  68.92870597 156.03234199 101.80734511
 108.60244399 214.13968395 218.58499151 131.09438025 114.99269899
 187.63688894 117.5399017  165.85959666 110.74981366 238.97897095
 114.01806217 240.07138204 283.0195311  108.08395213  67.44250127
 238.97271264]

Преимущества и недостатки LightGBM

Преимущества:

высокая точность прогнозов и особенно скорость обучения;
возможность работы с категориальными признаками;
большое количество гиперпараметров для настройки алгоритма;
применяется для многих задач.

Недостатки:

возможна нестабильная работа на датасетах небольшого размера;
сложность в конфигурации из-за большого количества гиперпараметров.

Дополнительные источники

Статья «LightGBM: A Highly Efficient Gradient Boosting Decision Tree», Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, Tie-Yan Liu.

Документация.

Лекции:

Какой бустинг лучше

Однозначного ответа на вопрос какой бустинг лучше нет. Если раньше можно было сказать, что при работе с категориальными признаками лучше использовать CatBoost, когда важна скорость — LightGBM, ну или XGBoost, если хочется стабильное решение и нет времени на эксперименты, то сейчас такое утверждение будет неверным: как можно было заметить, все три модификации успели скопировать друг у друга много хороших идей, но исторически они отличались довольно сильно.

Более правильным будет утверждение, что выбор наилучшего бустинга будет зависеть от конкретной задачи и данных, которые применяются для обучения модели: в разных сценариях одни и те же алгоритмы могут показывать как лучшие результаты в плане скорости и точности, так и наоборот — худшие. Таким образом, выбор наиболее оптимального бустинга будет заключаться в проведении экспериментов с разными модификациями и сравнении полученных результатов.

Несколько полезных источников со сравнением бустингов: один, два, три.

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13 мар. 2024 г.

06.09.24 01:44 Celinagarcia

HOW TO RECOVER MONEY LOST IN BITCOIN/USDT TRADING OR TO CRYPTO INVESTMENT !! Hi all, friends and families. I am writing From Alberton Canada. Last year I tried to invest in cryptocurrency trading in 2023, but lost a significant amount of money to scammers. I was cheated of my money, but thank God, I was referred to Hack Recovery Wizard they are among the best bitcoin recovery specialists on the planet. they helped me get every penny I lost to the scammers back to me with their forensic techniques. and I would like to take this opportunity to advise everyone to avoid making cryptocurrency investments online. If you have already lost money on forex, cryptocurrency or Ponzi schemes, please contact [email protected] or WhatsApp: +1 (757) 237–1724 at once they can help you get back the crypto you lost to scammers. BEST WISHES. Celina Garcia.
16.09.24 00:10 marcusaustin

Bitcoin Recovery Services: Restoring Lost Cryptocurrency If you've lost access to your cryptocurrency and unable to make a withdrawal, I highly recommend iBolt Cyber Hacker Bitcoin Recovery Services. Their team is skilled, professional, and efficient in recovering lost Bitcoin. They provide clear communication, maintain high security standards, and work quickly to resolve issues. Facing the stress of lost cryptocurrency, iBolt Cyber Hacker is a trusted service that will help you regain access to your funds securely and reliably. Highly recommended! Email: S u p p o r t @ ibolt cyber hack . com Cont/Whtp + 3. .9 .3. .5..0. .9. 2. 9. .0 .3. 1 .8. Website: h t t p s : / / ibolt cyber hack . com /
16.09.24 00:11 marcusaustin

Bitcoin Recovery Services: Restoring Lost Cryptocurrency If you've lost access to your cryptocurrency and unable to make a withdrawal, I highly recommend iBolt Cyber Hacker Bitcoin Recovery Services. Their team is skilled, professional, and efficient in recovering lost Bitcoin. They provide clear communication, maintain high security standards, and work quickly to resolve issues. Facing the stress of lost cryptocurrency, iBolt Cyber Hacker is a trusted service that will help you regain access to your funds securely and reliably. Highly recommended! Email: S u p p o r t @ ibolt cyber hack . com Cont/Whtp + 3. .9 .3. .5..0. .9. 2. 9. .0 .3. 1 .8. Website: h t t p s : / / ibolt cyber hack . com /
23.09.24 18:56 matthewshimself

At first, I was admittedly skeptical about Worldcoin (ref: https://worldcoin.org/blog/worldcoin/this-is-worldcoin-video-explainer-series), particularly around the use of biometric data and the WLD token as a reward mechanism for it. However, after following the project closer, I’ve come to appreciate the broader vision and see the value in the underlying tech behind it. The concept of Proof of Personhood (ref: https://worldcoin.org/blog/worldcoin/proof-of-personhood-what-it-is-why-its-needed) has definitely caught my attention, and does seem like a crucial step towards tackling growing issues like bots, deepfakes, and identity fraud. Sam Altman’s vision is nothing short of ambitious, but I do think he & Alex Blania have the chops to realize it as mainstay in the global economy.
01.10.24 14:54 Sinewclaudia

I lost about $876k few months ago trading on a fake binary option investment websites. I didn't knew they were fake until I tried to withdraw. Immediately, I realized these guys were fake. I contacted Sinew Claudia world recovery, my friend who has such experience before and was able to recover them, recommended me to contact them. I'm a living testimony of a successful recovery now. You can contact the legitimate recovery company below for help and assistance. [email protected] [email protected] WhatsApp: 6262645164
02.10.24 22:27 Emily Hunter

Can those who have fallen victim to fraud get their money back? Yes, you might be able to get back what was taken from you if you fell prey to a fraud from an unregulated investing platform or any other scam, but only if you report it to the relevant authorities. With the right plan and supporting documentation, you can get back what you've lost. Most likely, the individuals in control of these unregulated platforms would attempt to convince you that what happened to your money was a sad accident when, in fact, it was a highly skilled heist. You should be aware that there are resources out there to help you if you or someone you know has experienced one of these circumstances. Do a search using (deftrecoup (.) c o m). Do not let the perpetrators of this hoaxes get away with ruining you mentally and financially.
18.10.24 09:34 freidatollerud

The growth of WIN44 in Brazil is very interesting! If you're looking for more options for online betting and casino games, I recommend checking out Casinos in Brazil. It's a reliable platform that offers a wide variety of games and provides a safe and enjoyable experience for users. It's worth checking out! https://win44.vip
31.10.24 00:13 ytre89

Can those who have fallen victim to fraud get their money back? Yes, you might be able to get back what was taken from you if you fell prey to a fraud from an unregulated investing platform or any other scam, but only if you report it to the relevant authorities. With the right plan and supporting documentation, you can get back what you've lost. Most likely, the individuals in control of these unregulated platforms would attempt to convince you that what happened to your money was a sad accident when, in fact, it was a highly skilled heist. You should be aware that there are resources out there to help you if you or someone you know has experienced one of these circumstances. Do a search using (deftrecoup (.) c o m). Do not let the perpetrators of this hoaxes get away with ruining you mentally and financially.
02.11.24 14:44 diannamendoza732

In the world of Bitcoin recovery, Pro Wizard Gilbert truly represents the gold standard. My experience with Gilbert revealed just how exceptional his methods are and why he stands out as the premier authority in this critical field. When I first encountered the complexities of Bitcoin recovery, I was daunted by the technical challenges and potential risks. Gilbert’s approach immediately distinguished itself through its precision and effectiveness. His methods are meticulously designed, combining cutting-edge techniques with an in-depth understanding of the Bitcoin ecosystem. He tackled the recovery process with a level of expertise and thoroughness that was both impressive and reassuring. What sets Gilbert’s methods apart is not just their technical sophistication but also their strategic depth. He conducts a comprehensive analysis of each case, tailoring his approach to address the unique aspects of the situation. This personalized strategy ensures that every recovery effort is optimized for success. Gilbert’s transparent communication throughout the process was invaluable, providing clarity and confidence during each stage of the recovery. The results I achieved with Pro Wizard Gilbert’s methods were remarkable. His gold standard approach not only recovered my Bitcoin but did so with an efficiency and reliability that exceeded my expectations. His deep knowledge, innovative techniques, and unwavering commitment make him the definitive expert in Bitcoin recovery. For anyone seeking a benchmark in Bitcoin recovery solutions, Pro Wizard Gilbert’s methods are the epitome of excellence. His ability to blend technical prowess with strategic insight truly sets him apart in the industry. Call: for help. You may get in touch with them at ; Email: (prowizardgilbertrecovery(@)engineer.com) Telegram ; https://t.me/Pro_Wizard_Gilbert_Recovery Homepage ; https://prowizardgilbertrecovery.info
12.11.24 00:50 TERESA

Brigadia Tech Remikeable recovery has exhibited unparalleled strength in the realm of recovery. They stand out as the premier team to collaborate with if you encounter withdrawal difficulties from the platform where you’ve invested. Recently, I engaged with them to recover over a million dollars trapped in an investment platform I’d been involved with for months. I furnished their team with every detail of the investment, including accounts, names, and wallet addresses to which I sent the funds. This decision proved to be the best I’ve made, especially after realizing the company had scammed me. Brigadia Tech Remikeable recovery ensures exemplary service delivery and ensures the perpetrators face justice. They employ advanced techniques to ensure you regain access to your funds. Understandably, many individuals who have fallen victim to investment scams may still regret engaging in online services again due to the trauma of being scammed. However, I implore you to take action. Seek assistance from Brigadia Tech Remikeable Recovery Experts today and witness their remarkable capabilities. I am grateful that I resisted their enticements, and despite the time it took me to discover brigadia tech recovery, they ultimately fulfilled my primary objective. Without Brigadia Tech Recovery's intervention, I would have remained despondent and perplexed indefinitely. Also if you are looking for the best and safest investment company you can contact them, for wallet recovery, difficult withdrawal, etc. I am so happy to keep getting my daily BTC, all I do is keep 0.1 BTC in my mining wallet with the help of Brigadia Tech. They connected me to his mining stream and I earn 0.4 btc per day with this, my daily profit. I can get myself a new house and car. I can’t believe I have thousands of dollars in my bank account. Now you can get in. ([email protected]) Telegram +1 (323)-9 1 0 -1 6 0 5
17.11.24 09:31 Vivianlocke223

Have You Fallen Victim to Cryptocurrency Fraud? If your Bitcoin or other cryptocurrencies were stolen due to scams or fraudulent activities, Free Crypto Recovery Fixed is here to help you recover what’s rightfully yours. As a leading recovery service, we specialize in restoring lost cryptocurrency and assisting victims of fraud — no matter how long ago the incident occurred. Our experienced team leverages cutting-edge tools and expertise to trace and recover stolen assets, ensuring swift and secure results. Don’t let scammers jeopardize your financial security. With Free Crypto Recovery Fixed, you’re putting your trust in a reliable and dedicated team that prioritizes recovering your assets and ensuring their future protection. Take the First Step Toward Recovery Today! 📞 Text/Call: +1 407 212 7493 ✉️ Email: [email protected] 🌐 Website: https://freecryptorecovery.net Let us help you regain control of your financial future — swiftly and securely.
19.11.24 03:06 [email protected]

My entire existence fell apart when a malevolent hacker recently gained access to my online accounts. I felt violated and extremely uneasy after discovering that the digital platforms I depended on for communication, employment, and finances had been compromised. Regaining control and restoring my digital security was an overwhelming task in the immediate aftermath. To help me navigate the difficult process of recovering my accounts and getting my peace of mind back, TRUST GEEKS HACK EXPERT came into my life as a ray of hope. They immediately put their highly skilled professionals to work, thoroughly examining the vulnerability and methodically preventing unwanted access. They guided me through each stage soothingly, explaining what was occurring and why, so I never felt lost or alone. They communicated with service providers to restore my legitimate access while skillfully navigating the complex labyrinth of account recovery procedures. My digital footprint was cleaned and strengthened against future attacks thanks to their equally amazing ability to remove any remaining evidence of the hacker's presence. However, TRUST GEEKS HACK EXPERT actual worth went beyond its technical aspects. They offered constant emotional support during the ordeal, understanding my fragility and sense of violation. My tense nerves were calmed by their comforting presence and kind comments, which served as a reminder that I wasn't alone in this struggle. With their help, I was able to reestablish my sense of security and control, which enabled me to return my attention to the significant areas of my life that had been upended. Ultimately, TRUST GEEKS HACK EXPERT all-encompassing strategy not only recovered my online accounts but also my general peace of mind, which is a priceless result for which I am incredibly appreciative of their knowledge and kindness. Make the approach and send a message to TRUST GEEKS HACK EXPERT Via Web site <> www://trustgeekshackexpert.com/-- E>mail: Trustgeekshackexpert(At)fastservice..com -- TeleGram,<> Trustgeekshackexpert
19.11.24 03:07 [email protected]

My entire existence fell apart when a malevolent hacker recently gained access to my online accounts. I felt violated and extremely uneasy after discovering that the digital platforms I depended on for communication, employment, and finances had been compromised. Regaining control and restoring my digital security was an overwhelming task in the immediate aftermath. To help me navigate the difficult process of recovering my accounts and getting my peace of mind back, TRUST GEEKS HACK EXPERT came into my life as a ray of hope. They immediately put their highly skilled professionals to work, thoroughly examining the vulnerability and methodically preventing unwanted access. They guided me through each stage soothingly, explaining what was occurring and why, so I never felt lost or alone. They communicated with service providers to restore my legitimate access while skillfully navigating the complex labyrinth of account recovery procedures. My digital footprint was cleaned and strengthened against future attacks thanks to their equally amazing ability to remove any remaining evidence of the hacker's presence. However, TRUST GEEKS HACK EXPERT actual worth went beyond its technical aspects. They offered constant emotional support during the ordeal, understanding my fragility and sense of violation. My tense nerves were calmed by their comforting presence and kind comments, which served as a reminder that I wasn't alone in this struggle. With their help, I was able to reestablish my sense of security and control, which enabled me to return my attention to the significant areas of my life that had been upended. Ultimately, TRUST GEEKS HACK EXPERT all-encompassing strategy not only recovered my online accounts but also my general peace of mind, which is a priceless result for which I am incredibly appreciative of their knowledge and kindness. Make the approach and send a message to TRUST GEEKS HACK EXPERT Via Web site <> www://trustgeekshackexpert.com/-- E>mail: Trustgeekshackexpert(At)fastservice..com -- TeleGram,<> Trustgeekshackexpert
21.11.24 04:14 ronaldandre617

Being a parent is great until your toddler figures out how to use your devices. One afternoon, I left my phone unattended for just a few minutes rookie mistake of the century. I thought I’d take a quick break, but little did I know that my curious little genius was about to embark on a digital adventure. By the time I came back, I was greeted by two shocking revelations: my toddler had somehow managed to buy a $5 dinosaur toy online and, even more alarmingly, had locked me out of my cryptocurrency wallet holding a hefty $75,000. Yes, you heard that right a dinosaur toy was the least of my worries! At first, I laughed it off. I mean, what toddler doesn’t have a penchant for expensive toys? But then reality set in. I stared at my phone in disbelief, desperately trying to guess whatever random string of gibberish my toddler had typed as a new password. Was it “dinosaur”? Or perhaps “sippy cup”? I felt like I was in a bizarre game of Password Gone Wrong. Every attempt led to failure, and soon the laughter faded, replaced by sheer panic. I was in way over my head, and my heart raced as the countdown of time ticked away. That’s when I decided to take action and turned to Digital Tech Guard Recovery, hoping they could solve the mystery that was my toddler’s handiwork. I explained my predicament, half-expecting them to chuckle at my misfortune, but they were incredibly professional and empathetic. Their confidence put me at ease, and I knew I was in good hands. Contact With WhatsApp: +1 (443) 859 - 2886 Email digital tech guard . com Telegram: digital tech guard recovery . com website link :: https : // digital tech guard . com Their team took on the challenge like pros, employing their advanced techniques to unlock my wallet with a level of skill I can only describe as magical. As I paced around, anxiously waiting for updates, I imagined my toddler inadvertently locking away my life savings forever. But lo and behold, it didn’t take long for Digital Tech Guard Recovery to work their magic. Not only did they recover the $75,000, but they also gave me invaluable tips on securing my wallet better like not leaving it accessible to tiny fingers! Who knew parenting could lead to such dramatic situations? Crisis averted, and I learned my lesson: always keep my devices out of reach of little explorers. If you ever find yourself in a similar predicament whether it’s tech-savvy toddlers or other digital disasters don’t hesitate to reach out to Digital Tech Guard Recovery. They saved my funds and my sanity, proving that no challenge is too great, even when it involves a toddler’s mischievous fingers!
21.11.24 08:02 Emily Hunter

If I hadn't found a review online and filed a complaint via email to support@deftrecoup. com , the people behind this unregulated scheme would have gotten away with leaving me in financial ruins. It was truly the most difficult period of my life.
22.11.24 04:41 [email protected]

I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
22.11.24 15:26 cliftonhandyman

Your Lost Bitcoins Are Not Gone Forever? Enquire From iBolt Cyber Hacker iBolt Cyber Hacker is a cybersecurity service that specializes in Bitcoin and cryptocurrency recovery. Even if your Bitcoin is locked away in a scammer inaccessible wallet, they have the tools and expertise to retrieve it. Many people, including seasoned cryptocurrency investors, face the daunting possibility of never seeing their lost funds again. iBolt cyber hacker service is a potential lifeline in these situations. I understand the concerns many people might have about trusting a third-party service to recover their Bitcoin. iBolt Cyber Hacker takes security seriously, implementing encryption and stringent privacy protocols. I was assured that no sensitive data would be compromised during the recovery process. Furthermore, their reputation in the cryptocurrency community, based on positive feedback from previous clients, gave me confidence that I was in good hands. Whtp +39, 351..105, 3619 Em.ail: ibolt @ cyber- wizard. co m
22.11.24 23:43 teresaborja

all thanks to Tech Cyber Force Recovery expert assistance. As a novice in cryptocurrency, I had been carefully accumulating a modest amount of Bitcoin, meticulously safeguarding my digital wallet and private keys. However, as the adage goes, the best-laid plans can often go awry, and that's precisely what happened to me. Due to a series of technical mishaps and human errors, I found myself locked out of my Bitcoin wallet, unable to access the fruits of my digital labors. Panic set in as I frantically searched for a solution, scouring the internet for any glimmer of hope. That's when I stumbled upon the Tech Cyber Force Recovery team, a group of seasoned cryptocurrency specialists who had built a reputation for their ability to recover lost or inaccessible digital assets. Skeptical at first, I reached out, desperate for a miracle. To my utter amazement, the Tech Cyber Force Recovery experts quickly assessed my situation and devised a meticulous plan of attack. Through their deep technical knowledge, unwavering determination, and a keen eye for detail, they were able to navigate the complex labyrinth of blockchain technology, ultimately recovering my entire Bitcoin portfolio. What had once seemed like a hopeless endeavor was now a reality, and I found myself once again in possession of my digital wealth, all thanks to the incredible efforts of the Tech Cyber Force Recovery team. This experience has not only restored my faith in the cryptocurrency ecosystem. Still, it has also instilled in me a profound appreciation for the critical role that expert recovery services can play in safeguarding one's digital assets. ENAIL < Tech cybers force recovery @ cyber services. com > WEBSITE < ht tps : // tech cyber force recovery. info > TEXT < +1. 561. 726. 3697 >
24.11.24 02:21 [email protected]

I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
25.11.24 02:19 briankennedy

COMMENT ON I NEED A HACKER TO RECOVER MONEY FROM BINARY TRADING. HIRE FASTFUND RECOVERY
25.11.24 02:20 briankennedy

After countless hours of research and desperate attempts to find a solution, I stumbled upon FASTFUND RECOVERY. It was like finding an oasis in the middle of a desert. Their website promised to help victims of scams reclaim what was rightfully theirs, and I instantly knew I had to give them a shot. Before diving headfirst into the recovery process, I wanted to make sure that FASTFUND RECOVERY was the real deal. So, I did my due diligence and looked into their expertise and reputation. To my relief, I found that they had an impeccable track record, successfully assisting countless individuals in recovering their lost funds. Their team consisted of experts in cybersecurity and financial fraud, armed with the knowledge and tools needed to tackle even the most intricate scams. With their reputation preceding them, I felt a renewed sense of hope. FASTFUND RECOVERY successfully came to my aid and got back the amount I lost to these scammers and for this, I am sending this article for clarification. The info of FASTFUND RECOVERY is email: Fastfundrecovery8 (@)Gmail (.) com. Web fastfundrecovery(.)com. (W/A 1 807/500/7554)
26.11.24 21:59 [email protected]

In a world brimming with enticing investment opportunities, it is crucial to tread carefully. The rise of digital currencies has attracted many eager investors, but along with this excitement lurk deceitful characters ready to exploit the unsuspecting. I learned this lesson the hard way, and I want to share my story in the hopes that it can save someone from making the same mistakes I did. It all began innocently enough when I came across an engaging individual on Facebook. Lured in by promises of high returns in the cryptocurrency market, I felt the electric thrill of potential wealth coursing through me. Initial investments returned some profits, and that exhilarating taste of success fueled my ambition. Encouraged by a meager withdrawal, I decided to commit even more funds. This was the moment I let my guard down, blinded by greed. As time went on, the red flags started to multiply. The moment I tried to withdraw my earnings, a cascade of unreasonable fees appeared like a thick mist, obscuring the truth. “Just a little more,” they said, “Just until the next phase.” I watched my hard-earned money slip through my fingers as I scraped together every last cent to pay those relentless fees. My trust had become my downfall. In the end, I lost not just a significant amount of cash, but my peace of mind about $1.1 million vanished into the abyss of false promises and hollow guarantees. But despair birthed hope. After a cascade of letdowns, I enlisted the help of KAY-NINE CYBER SERVICES, a team that specializes in reclaiming lost funds from scams. Amazingly, they worked tirelessly to piece together what had been ripped away, providing me with honest guidance when I felt utterly defeated. Their expertise in navigating the treacherous waters of crypto recovery was a lifeline I desperately needed. To anyone reading this, please let my story serve as a warning. High returns often come wrapped in the guise of deception. Protect your investments, scrutinize every opportunity, and trust your instincts. Remember, the allure of quick riches can lead you straight to heartbreak, but with cautious determination and support, it is possible to begin healing from such devastating loss. Stay informed, stay vigilant, and may you choose your investment paths wisely. Email: kaynine @ cyberservices . com
26.11.24 23:12 rickrobinson8

FAST SOLUTION FOR CYPTOCURRENCY RECOVERY SPARTAN TECH GROUP RETRIEVAL
26.11.24 23:12 rickrobinson8

Although recovering from the terrible effects of investment fraud can seem like an impossible task, it is possible to regain financial stability and go on with the correct assistance and tools. In my own experience with Wizard Web Recovery, a specialized company that assisted me in navigating the difficulties of recouping my losses following my fall prey to a sophisticated online fraud, that was undoubtedly the case. My life money had disappeared in an instant, leaving me in a state of shock when I first contacted Spartan Tech Group Retrieval through this Email: spartantechretrieval (@) g r o u p m a i l .c o m The compassionate and knowledgeable team there quickly put my mind at ease, outlining a clear and comprehensive plan of action. They painstakingly examined every aspect of my case, using their broad business contacts and knowledge to track the movement of my pilfered money. They empowered me to make knowledgeable decisions regarding the rehabilitation process by keeping me updated and involved at every stage. But what I valued most was their unrelenting commitment and perseverance; they persisted in trying every option until a sizable amount of my lost money had been successfully restored. It was a long and arduous journey, filled with ups and downs, but having Spartan Tech Group Retrieval in my corner made all the difference. Thanks to their tireless efforts, I was eventually able to rebuild my financial foundation and reclaim a sense of security and control over my life. While the emotional scars of investment fraud may never fully heal, working with this remarkable organization played a crucial role in my ability to move forward and recover. For proper talks, contact on WhatsApp:+1 (971) 4 8 7 - 3 5 3 8 and Telegram:+1 (581) 2 8 6 - 8 0 9 2 Thank you for your time reading as it will be of help.
27.11.24 00:39 [email protected]

Although recovering lost or inaccessible Bitcoin can be difficult and unpleasant, it is frequently possible to get back access to one's digital assets with the correct help and direction. Regarding the subject at hand, the examination of Trust Geeks Hack Expert Website www://trustgeekshackexpert.com/ assistance after an error emphasizes how important specialized services may be in negotiating the difficulties of Bitcoin recovery. These providers possess the technical expertise and resources necessary to assess the situation, identify the root cause of the issue, and devise a tailored solution to retrieve the lost funds. By delving deeper into the specifics of Trust Geeks Hack Expert approach, we can gain valuable insights into the nuances of this process. Perhaps they leveraged advanced blockchain analysis tools to trace the transaction history and pinpoint the location of the missing Bitcoins. Or they may have collaborated with the relevant parties, such as exchanges or wallet providers, to facilitate the recovery process. Equally important is the level of personalized support and communication that Trust Geeks Hack Expert likely provided, guiding the affected individual through each step of the recovery effort and offering reassurance during what can be an anxious and uncertain time. The success of their efforts, as evidenced by the positive outcome, underscores the importance of seeking out reputable and experienced service providers when faced with a Bitcoin-related mishap, as they possess the specialized knowledge and resources to navigate these challenges and restore access to one's digital assets. Email.. [email protected]
27.11.24 09:10 Michal Novotny

The biggest issue with cryptocurrency is that it is unregulated, wh ich is why different people can come up with different fake stories all the time, and it is unfortunate that platforms like Facebook and others only care about the money they make from them through ads. I saw an ad on Facebook for Cointiger and fell into the scam, losing over $30,000. I reported it to Facebook, but they did nothing until I discovered deftrecoup . c o m from a crypto community; they retrieved approximately 95% of the total amount I lost.
01.12.24 17:21 KollanderMurdasanu

REACH OUT TO THEM WhatsApp + 156 172 63 697 Telegram (@)Techcyberforc We were in quite a bit of distress. The thrill of our crypto investments, which had once sparked excitement in our lives, was slowly turning into anxiety when my husband pointed out unusual withdrawal issues. At first, we brushed it off as minor glitches, but the situation escalated when we found ourselves facing login re-validation requests that essentially locked us out of our crypto wallet—despite entering the correct credentials. Frustrated and anxious, we sought advice from a few friends, only to hit a wall of uncertainty. Turning to the vast expanse of the internet felt daunting, but in doing so, we stumbled upon TECH CYBER FORCE RECOVERY. I approached them with a mix of skepticism and hope; after all, my understanding of these technical matters was quite limited. Yet, from our very first interaction, it was clear that they were the experts we desperately needed. They walked us through the intricacies of the recovery process, patiently explaining each mechanism—even if some of it went over my head, their reassurance was calming. Our responsibility was simple: to provide the correct information to prove our ownership of the crypto account, and thankfully, we remained on point in our responses. in a timely fashion, TECH CYBER FORCE RECOVERY delivered on their promises, addressing all our withdrawal and access issues exactly when they said they would. The relief we felt was immense, and the integrity they displayed made me confident in fully recommending their services. If you ever find yourself in a similar predicament with your crypto investments, I wholeheartedly suggest reaching out to them. You can connect with TECH CYBER FORCE RECOVERY through their contact details for assistance and valuable guidance. Remember, hope is only a reach away!
02.12.24 23:02 ytre89

Online crypto investment can seem like a promising opportunity, but it's crucial to recognize that there are no guarantees. My experience serves as a stark reminder of this reality. I was drawn in by the allure of high returns and the persuasive marketing tactics employed by various brokers. Their polished presentations and testimonials made it seem easy to profit from cryptocurrency trading. Everything appeared to be legitimate. I received enticing messages about the potential for substantial gains, and the brokers seemed knowledgeable and professional. Driven by excitement and the fear of missing out, I invested a significant amount of my savings. The promise of quick profits overshadowed the red flags I should have noticed. I trusted these brokers without conducting proper research, which was a major mistake. As time went on, I realized that the promised returns were nothing but illusions. My attempts to withdraw funds were met with endless excuses and delays. It became painfully clear that I had fallen victim. The reality hit hard: my hard-earned money was gone, I lost my peace of mind and sanity. In my desperation, I sought help from a company called DEFTRECOUP. That was the turning point for me as I had a good conversation and eventually filed a complaint via DEFTRECOUP COM. They were quite delicate and ensured I got out of the most difficult situation of my life in one piece.
04.12.24 22:24 andreygagloev

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12.12.24 00:35 amandagregory

HOW TO HIRE A HACKER TO RECOVER STOLEN BITCOIN WITH FASTFUND RECOVERY
12.12.24 00:35 amandagregory

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27.12.24 00:21 swiftdream

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06.01.25 19:09 michaeljordan15

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18.01.25 12:41 michaeldavenport218

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After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
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After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
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26.01.25 03:54 [email protected]

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02.02.25 20:53 Michael9090

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06.02.25 19:42 Marta Golomb

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08.02.25 05:45 [email protected]

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08.02.25 05:46 [email protected]

I'm incredibly grateful that I did enough research to recover my stolen cryptocurrency. When I first fell victim to a scam, I felt hopeless and lost, unsure if I'd ever see my funds again. A few months ago, I was approached by someone on Telegram who claimed to have a lucrative investment opportunity in cryptocurrencies. They promised huge returns and played on my emotions, making it seem like a can't-miss chance. I was so eager to make my money grow that I didn't fully vet the situation, and unfortunately, I ended up falling for the scam. They guided me to invest a significant amount of money, and soon after, I realized I had been duped. The scammers blocked me, and my funds were gone. I felt devastated. All of my savings had been wiped out in what seemed like an instant, and the feeling of being taken advantage of was crushing. I spent days researching how to recover my stolen cryptocurrency but found the process to be overwhelming and complicated. I was starting to lose hope when I came across Trust Geeks Hack Expert. At first, I was skeptical about reaching out to a cryptocurrency recovery company, but after reading testimonials and researching their reputation, I decided to give them a try. I contacted Trust Geeks Hack Expert Website: www://trustgeekshackexpert.com/, and I was immediately reassured by their professionalism and expertise. They took the time to listen to my situation, and they were honest about what could and could not be done. What stood out to me was their deep understanding of cryptocurrency fraud and the recovery process. They were able to track down the scammers and initiate the recovery of my stolen funds, step by step. Thanks to Trust Geeks Hack Expert, I was able to get back a significant portion of the cryptocurrency I had lost. Their team was responsive, transparent, and diligent in their efforts. I was kept informed throughout the entire process, and they made sure I felt supported every step of the way. I truly can't thank them enough for their dedication and for restoring my faith in the possibility of recovery after such a devastating loss. I will definitely recommend Trust Geeks Hack Expert to anyone who has fallen victim to a cryptocurrency scam. TeleGram: Trustgeekshackexpert & what's A p p +1 7 1 9 4 9 2 2 6 9 3
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11.02.25 04:24 heyemiliohutchinson

I invested substantially in Bitcoin, believing it would secure my future. For a while, things seemed to be going well. The market fluctuated, but I was confident my investment would pay off. But catastrophe struck without warning. I lost access to my Bitcoin holdings as a result of several technical issues and inadequate security measures. Every coin in my wallet suddenly disappeared, leaving me with an overpowering sense of grief. The emotional impact of this loss was far greater than I had imagined. I spiraled into despair, feeling as though my dreams of financial independence were crushed. I was on the verge of giving up when I came across Assets_Recovery_Crusader. Being willing to give them a chance, I had nothing left to lose. They listened to my narrative and took the time to comprehend the particulars of my circumstance, rather than treating me like a case number. They worked diligently, using their advanced recovery techniques and deep understanding of blockchain technology to track down my lost Bitcoin. Assets_Recovery_Crusader rebuilt my trust in the bitcoin space. The financial impact had a significant emotional toll, but I was able to get past it thanks to Assets_Recovery_Crusader’s proficiency and persistence. For proper talks, reach out to them via TELEGRAM : Assets_Recovery_Crusader EMAIL: [email protected]
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HIRE A HACKE DUNAMIS CYBER SOLUTION
11.02.25 22:46 jimmybrown

HIRE A HACKE DUNAMIS CYBER SOLUTIONI was just hours away from sealing the biggest real estate deal of my life- the kind of deal that would make one feel like a financial genius. It was the dream property, and all I had to do was transfer my $450,000 Bitcoin deposit. Simple, right? Wrong. I pulled up my crypto wallet, ready to finalize the transfer, and access was denied. No big deal. Maybe I mistyped the password. I tried again. Access denied. Panic started seeping in. I switched devices. Rebooted my system. I entered every password I had ever used since the dawn of time, including my childhood nickname and my favorite pizza topping. Still. Nothing. This wasn't a glitch. This was a full-scale disaster. The seller was waiting; my real estate agent was waiting. And my money? Trapped in a digital vault I suddenly had no key to. Every worst-case scenario flooded my head: Had I been hacked? Did I lock myself out? Was this some kind of cosmic payback for every time I blew off software updates? Just about the time I was getting comfortable in my new identity as the guy who almost bought a house, I remembered that a friend, a crypto lawyer-once said something to me about a recovery service. I called him with the urgency of a man dangling off a cliff. The moment I said what happened, he cut me off: "Email DUNAMIS CYBER SOLUTION Recovery. Now." I didn't ask questions. I dialed quicker than I'd ever dialed in my life. From the second they answered, I knew I was with the pros. There was no hemming, no hawing; this team must have handled its fair share of this particular type of nightmare. They talked me through the process, asked the right questions, and went to work like surgeons in a digital operating room. Minutes felt like hours. I was at DEFCON 1, stress-wise. I paced and stared at my phone, wondering if it was time to move into a cave because, at this rate, homeownership was not looking good. Then—the call came: "We got it." I just about collapsed with relief. My funds were safe. My wallet was unlocked. The Bitcoin was transferred just in time, and I signed the contract with literal seconds to spare. And that night, almost lost to the tech catastrophe of the century in that house, I made a couple of vows: never underestimate proper wallet management and always keep DUNAMIS CYBER SOLUTION Recovery on speed dial. [email protected] +13433030545
13.02.25 14:45 aoifewalsh130

TELEGRAM: u/BestwebwizardRecovery EMAIL: [email protected] WEBSITE: https://bestwebwizrecovery.com/ The money I invested was meant for something incredibly important—my wedding. After months of saving, I had finally accumulated enough to make the day truly special. Wanting to grow this fund, I came across a crypto site called Abcfxb.pro, which promised daily returns through “AI crypto arbitrage trading.” They claimed they could deliver 1% returns on my investment every day, and I saw this as an opportunity to multiply my savings quickly. I thought it was the perfect way to ensure I’d have enough to cover all the wedding expenses. For the first few days, everything seemed perfect. I saw the promised returns and was able to withdraw money without any issues. It felt like a legitimate opportunity, and I was excited as my wedding fund grew. However, things took a turn when I tried to withdraw again. The site claimed that my account balance had fallen below their liquidity requirement and asked me to deposit more funds to proceed. Reluctantly, I deposited more money, believing it was just a minor issue. But the situation only worsened. I was then told that my withdrawal would take 50 days due to “blockchain congestion.” I wasn’t too concerned at first, thinking it was just a delay. But after 50 days, I still hadn’t received my funds, and they gave me the same excuse. Desperate, I contacted the site again, only to be informed that I would need to pay a 15% fee for “technical support” from the “Federal Reserve’s blockchain regulator” before I could withdraw my money. By now, I realized I had fallen victim to a scam. As I researched further, I found that others had been scammed in the same way, and the scammers had moved to another site with nearly the same layout. It was then that I came across a review from another victim, who explained how Best Web Wizard Recovery had helped him recover his lost funds. Desperate for a solution, I reached out to Best Web Wizard Recovery. To my relief, they responded quickly and professionally. Within six hours, they had successfully recovered my full investment. I was beyond grateful, especially since the money had been intended for my wedding. Thanks to their help, I was able to not only get my money back but also go ahead with my wedding as planned. It was a day I will always cherish, and I owe it to Best Web Wizard Recovery for helping me make it a reality. I highly recommend their services to anyone who has fallen victim to a crypto scam.
13.02.25 16:50 andytom798

I lost $210,000 worth of Bitcoin to a group of fake blockchain impostors on Red note, a Chinese app. They contacted me, pretending to be official blockchain support, and I was misled into believing they were legitimate. At the time, I had been saving up in Bitcoin, hoping to take advantage of the rising market. The scammers were convincing, and I made the mistake of trusting them with access to my blockchain wallet. To my shock and disbelief, they stole a total of $10,000 worth of Bitcoin from my wallet. It was devastating, as this amount represented all of my hard-earned savings. I was in utter disbelief, feeling foolish for falling for their deceptive tactics. I felt lost, as though everything I had worked towards was taken from me in an instant. Thankfully, my uncle suggested I reach out to an expert in cryptocurrency recovery. After doing some research online, I came across CYBERPOINT RECOVERY COMPANY. I was hesitant at first, but their positive reviews gave me some hope. I decided to contact them directly and explained my situation, including the amount I had lost and how the scammers had gained access to my account. To my relief, the team at CYBERPOINT RECOVERY responded quickly and assured me they could help. They launched a detailed recovery program, using advanced tools and techniques to trace the stolen Bitcoin. Within a matter of days, they successfully recovered my full $210,000 worth of Bitcoin, and they even identified the individuals behind the scam. Their expertise and professionalism made a huge difference, and I was incredibly grateful for their support. If you find yourself in a similar situation, I highly recommend reaching out to Cyber Constable Intelligence. They helped me recover my funds when I thought all hope was lost. Whether you’ve lost money to scammers or any other form of online fraud, they have the knowledge and resources to help you get your funds back. Don’t give up there are experts who can help you reclaim what you’ve lost. I’m sharing my story to hopefully guide others who are going through something similar. Here's Their Info Below ([email protected]) or W.H.A.T.S.A.P.P:+1.7.6.0.9.2.3.7.4.0.7
13.02.25 16:52 birenderkumar20101

I was able to reclaim my lost Bitcoin assets worth of $480,99 which i had lost to the scam company known as Capitalix fx a scam company pretending to be an investment platform which alot of people including myself have lost their funds to, sadly not all would be fortunate enough to retrieve back their funds like I did but if you’re reading this today then you’re already a step closer towards regaining your lost digital assets, CYBERPOINT RECOVERY COMPANY successfully retrieved back my funds in less than of 48hours after I sought for their help to get back my funds. This experience has taught me the importance of carrying out my due diligence before embracing any financial opportunity presented to me and while risk taking may be a part of the journey, some risks are not worth taking and never again will I involve myself with any online financial investment. It’s only right that we seek for external intervention and support of a higher knowledge system when it comes to digital assets recovery, Get in contact today with the team to get started on Email: ([email protected])
13.02.25 17:37 eunice49954

Agent Jasmine Lopez focuses on recovering stolen cryptocurrency, particularly USDT. She is well-known for helping victims of digital asset theft. Her reputation arises from successful recoveries that have allowed many to regain their lost funds. I witnessed this when $122,000 was taken from me. Thanks to Ms. Lopez's skills, I recovered the entire amount in just 24 hours. Her prompt response and effective methods relieved my financial burden. Ms. Lopez’s commitment to helping others is evident. She is always available to offer solutions to those facing similar problems. For assistance, she can be reached via email at recoveryfundprovider@gmail . com or contacted directly on WhatsApp and text at +44 - 7366 445035. Her Instagram handle is recoveryfundprovider.
14.02.25 02:50 Vladimir876

I was able to reclaim my lost Bitcoin assets worth of $480,99 which i had lost to the scam company known as Capitalix fx a scam company pretending to be an investment platform which alot of people including myself have lost their funds to, sadly not all would be fortunate enough to retrieve back their funds like I did but if you’re reading this today then you’re already a step closer towards regaining your lost digital assets, CYBERPOINT RECOVERY COMPANY successfully retrieved back my funds in less than of 48hours after I sought for their help to get back my funds. This experience has taught me the importance of carrying out my due diligence before embracing any financial opportunity presented to me and while risk taking may be a part of the journey, some risks are not worth taking and never again will I involve myself with any online financial investment. It’s only right that we seek for external intervention and support of a higher knowledge system when it comes to digital assets recovery, Get in contact today with the team to get started on Email: ([email protected]) or W.H.A.T.S.A.P.P:+1.7.6.0.9.2.3.7.4.0.7
14.02.25 02:56 christophadelbert3

Я был в полном смятении, когда потерял все свои сбережения, инвестируя в криптовалюту. Со мной связалась онлайн женщина по электронной почте, выдавая себя за менеджера по работе с клиентами банка, которая сказала мне, что я могу удвоить свои сбережения, инвестируя в криптовалюту. Я никогда не думал, что это будет мошенничество, и я потеряю все. Это продолжалось неделями, пока я не понял, что меня обманули. Вся надежда была потеряна, я был опустошен и разорен, к счастью для меня, я наткнулся на статью в моем местном бюллетене о CYBERPUNK RECOVERY Bitcoin Recovery. Я связался с ними и предоставил всю информацию по моему делу. Я был поражен тем, как быстро они вернули мои криптовалютные средства и смогли отследить этих мошенников. Я действительно благодарен за их услуги и рекомендую CYBERPUNK RECOVERY всем, кому нужно вернуть свои средства. Настоятельно рекомендую вам связаться с CYBERPUNK, если вы потеряли свои биткойны USDT или ETH из-за инвестиций в биткойны Электронная почта: ([email protected]) W.h.a.t.s.A.p.p (+.1.7.6.0.9.2.3.7.4.0.7)
14.02.25 02:56 christophadelbert3

I was in total dismay when I lost my entire savings investing in cryptocurrency, I was contacted online by a lady through email pretending to be an account manager of a bank, who told me I could make double my savings through cryptocurrency investment, I never imagined it would be a scam and I was going to lose everything. It went on for weeks until I realized that I have been scammed. All hope was lost, I was devastated and broke, fortunately for me, I came across an article on my local bulletin about CYBERPUNK RECOVERY Bitcoin Recovery, I contacted them and provided all the information regarding my case, I was amazed at how quickly they recovered my cryptocurrency funds and was able to trace down those scammers. I’m truly grateful for their service and I recommend CYBERPUNK RECOVERY to everyone who needs to recover their funds urge you to contact CYBERPUNK if you have lost your bitcoin USDT or ETH through bitcoin investment Email: ([email protected]) WhatsApp (+17609237407)
14.02.25 15:33 prelogmilivoj

I never imagined I would find myself in a situation where I was scammed out of such a significant amount of money, but it happened. I became a victim of a fake online donation project that cost me over $30,000. It all started innocently enough when I was searching for assistance after a devastating fire incident in California. While looking for support, I came across an advertisement that seemed to offer donations for fire victims. The ad appeared legitimate, and I reached out to the project manager to inquire about how to receive the donations. The manager was very convincing and insisted that in order to qualify for the donations, I needed to pay $30,000 upfront. In return, I was promised $1 million in donations. It sounded a bit too good to be true, but in my desperate situation, I made the mistake of believing it. The thought of receiving a substantial amount of help to rebuild after the fire clouded my judgment, and I went ahead and sent the money. However, after transferring the funds, the promised donations never arrived, and the manager disappeared. That’s when I realized I had been scammed. Feeling lost, helpless, and completely betrayed, I tried everything I could to contact the scammer, but all my efforts were in vain. Desperation led me to search for help online, hoping to find a way to recover my money and potentially track down the scammer. That’s when I stumbled upon several testimonies from others who had fallen victim to similar scams and had been helped by a company called Tech Cyber Force Recovery. I reached out to them immediately, providing all the details of the scam and the information I had gathered. To my immense relief, the experts at Tech Cyber Force Recovery acted swiftly. Within just 27 hours, they were able to locate the scammer and initiate the recovery process. Not only did they help me recover the $30,000 I had lost, but the most satisfying part was that the scammer was apprehended by local authorities in their region. Thanks to Tech Cyber Force Recovery, I was able to get my money back and hold the scammer accountable for their actions. I am incredibly grateful for their professionalism, expertise, and dedication to helping victims like me. If you have fallen victim to a scam or fraudulent activity, I highly recommend contacting Tech Cyber Force Recovery. They provide swift and efficient recovery assistance, and I can confidently say they made all the difference in my situation. ☎☎ 1️⃣5️⃣6️⃣1️⃣7️⃣2️⃣6️⃣3️⃣6️⃣9️⃣7️⃣ ☎☎ 📩 1️⃣5️⃣6️⃣1️⃣7️⃣2️⃣6️⃣3️⃣6️⃣9️⃣7️⃣ 📩
14.02.25 22:12 eunice49954

Agent Jasmine Lopez focuses on recovering stolen cryptocurrency, particularly USDT. She is well-known for helping victims of digital asset theft. Her reputation arises from successful recoveries that have allowed many to regain their lost funds. I witnessed this when $122,000 was taken from me. Thanks to Ms. Lopez's skills, I recovered the entire amount in just 24 hours. Her prompt response and effective methods relieved my financial burden. Ms. Lopez’s commitment to helping others is evident. She is always available to offer solutions to those facing similar problems. For assistance, she can be reached via email at recoveryfundprovider@gmail . com or contacted directly on WhatsApp and text at +44 - 7366 445035. Her Instagram handle is recoveryfundprovider.
15.02.25 02:51 Michelle Lynn

Living in Los Angeles, I never imagined I’d face such a difficult chapter in my life. At the time, my wife was pregnant, and we were both excited about starting a family. I fell victim to a series of scams, losing over $170,000 in total. Just when I thought things couldn’t get worse, I received a call from someone who promised to help me recover my losses. Desperate to fix the situation, I went along with it, hoping for a breakthrough. But it turned out to be another scam. However, most of the options I found either seemed dubious or offered no real guarantees. That’s when I came across Cyber Constable Intelligence. It was a company recommended in a Facebook community The team worked tirelessly on my case, and after some time, they successfully recovered 99% of my investment. Although I didn’t recover everything, the 99% recovery was a huge relief They also educated me on how to better protect my digital Asset Here's Their Website Info www cyberconstableintelligence com
15.02.25 02:51 Michelle Lynn

Living in Los Angeles, I never imagined I’d face such a difficult chapter in my life. At the time, my wife was pregnant, and we were both excited about starting a family. I fell victim to a series of scams, losing over $170,000 in total. Just when I thought things couldn’t get worse, I received a call from someone who promised to help me recover my losses. Desperate to fix the situation, I went along with it, hoping for a breakthrough. But it turned out to be another scam. However, most of the options I found either seemed dubious or offered no real guarantees. That’s when I came across Cyber Constable Intelligence. It was a company recommended in a Facebook community The team worked tirelessly on my case, and after some time, they successfully recovered 99% of my investment. Although I didn’t recover everything, the 99% recovery was a huge relief They also educated me on how to better protect my digital Asset Here's Their Website Info www cyberconstableintelligence com, WhatsApp Info: 1 (252) 378-7611
16.02.25 01:01 Peter

I fell victim to a crypto scam and lost a significant amount of money. What are the most effective strategies to recover my funds? I've heard about legal actions, contacting authorities, and hiring recovery experts, but I'm not sure where to start. Can you provide some guidance on the best ways to recover money lost in a crypto scam? Well if this is you, [email protected] gat you covered get in touch and thank me later
16.02.25 20:06 eunice49954

Agent Lopez specializes in recovering stolen cryptocurrencies, especially Bitcoin/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
18.02.25 19:35 donovancristina

Now, I’m that person sharing my success story on LinkedIn, telling others about the amazing team at TECH CYBER FORCE RECOVERY who literally saved my financial life. I’ve also become that guy who proudly shares advice like “Always back up your wallet, and if you don’t have TECH CYBER FORCE RECOVERY on speed dial.” So, a big thank you to TECH CYBER FORCE RECOVERY if I ever get a chance to meet the team, I might just offer to buy them a drink. They’ve earned it. FOR CRYPTO HIRING WEBSITE WWW://techcyberforcerecovery.com WHATSAPP : ⏩ wa.me/15617263697
18.02.25 22:13 keithphillip671

WhatsApp +44,7,4,9,3,5,1,3,3,8,5 Telegram @Franciscohack The day my son uncovered the truth—that the man I entrusted my hopes of wealth and companionship with through a cryptocurrency platform was a cunning scammer was the day my world crumbled. The staggering realization that I had been swindled out of 150,000.00 Euro worth of Bitcoin left me in a state of profound despair. As a 73-year-old grappling with loneliness, I had sought solace in what I believed to be a genuine connection, only to find deceit and betrayal. Countless sleepless nights were spent in tears, mourning not only the financial devastation but also the crushing blow to my trust. Attempts to verify the authenticity of our interactions were met with hostility, further deepening my sense of isolation. Through the loss it was my son who became my beacon of resilience. He took upon himself the arduous task of tracing the scam and seeking justice on my behalf. Through meticulous effort and determination, he unearthed {F R A N C I S C O H A C K}, renowned for their expertise in recovering funds lost to cryptocurrency scams. Entrusting them with screenshots and evidence of the fraudulent transactions, my son initiated the journey to reclaim what had been callously taken from me. {F R A N C I S C O H A C K} approached our plight with empathy and unwavering professionalism, immediately instilling a sense of confidence in their abilities. Despite my initial skepticism, their transparent communication and methodical approach reassured us throughout the recovery process. Regular updates on their progress and insights into their strategies provided much-needed reassurance and kept our hopes alive amid the uncertainty. Their commitment to transparency and client welfare was evident in every interaction, fostering a sense of partnership rather than mere service. Miraculously, in what felt like an eternity but was actually an impressively brief period, {F R A N C I S C O H A C K} delivered the astonishing news—I had recovered the entire 150,000.00 Euro worth of stolen Bitcoin. The flood of relief and disbelief was overwhelming, marking not just the restitution of financial losses but the restoration of my faith in justice. {F R A N C I S C O H A C K} proficiency in navigating the intricate landscape of blockchain technology and online fraud was nothing short of extraordinary. Their dedication to securing justice and restoring client confidence set them apart as more than just experts—they were steadfast allies in a fight against digital deceit. What resonated deeply with me {F R A N C I S C O H A C K} integrity and compassion. Despite the monumental recovery, they maintained transparency regarding their fees and ensured fairness in all dealings. Their proactive guidance on cybersecurity measures further underscored their commitment to safeguarding clients from future threats. It was clear that their mission extended beyond recovery—it encompassed education, prevention, and genuine advocacy for those ensnared by cyber fraud. ([email protected]) fills me with profound gratitude. They not only rescued my financial security but also provided invaluable emotional support during a time of profound vulnerability. To anyone navigating the aftermath of cryptocurrency fraud, I wholeheartedly endorse {F R A N C I S C O H A C K}. They epitomize integrity, expertise, and unwavering dedication to their clients' well-being. My experience with {F R A N C I S C O H A C K} transcended mere recovery—it was a transformative journey of resilience, restoration, and renewed hope in the face of adversity.
21.02.25 07:42 daniel231101

I never thought I would fall victim to a crypto scam until I was convinced of a crypto investment scam that saw me lose all my entire assets worth $487,000 to a crypto investment manager who convinced me I could earn more from my investment. I thought it was all gone for good but I kept looking for ways to get back my stolen crypto assets and finally came across Ethical Hack Recovery, a crypto recovery/spying company that has been very successful in the recovery of crypto for many other victims of crypto scams and people who lost access to their crypto. I’m truly grateful for their help as I was able to recover my stolen crypto assets and get my life back together. I highly recommend their services EMAIL ETHICALHACKERS009 AT @GMAIL DOT COM whatsapp +14106350697
21.02.25 21:38 eunice49954

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
22.02.25 18:01 benluna0991

Mark Zuckerberg. That’s the name I was introduced to when I first encountered the cryptocurrency mining platform, WHATS Invest. A person claiming to be Zuckerberg himself reached out to me, saying that he was personally backing the platform to help investors like me earn passive income. At first, I was skeptical—after all, how often do you get a direct connection to one of the world’s most famous tech entrepreneurs? But this individual seemed convincing and assured me that many people were already seeing substantial returns on their investments. He promised me a great opportunity to secure my financial future, so I decided to take the plunge and invest $10,000 into WHATS Invest. They told me that I could expect to see significant returns in just a few months, with payouts of at least $1,500 or more each month. I was excited, believing this would be my way out of financial struggles. However, as time passed, things didn’t go according to plan. Months went by, and I received very little communication. When I finally did receive a payout, it was nowhere near the $1,500 I was promised. Instead, I received just $200, barely 13% of what I had expected. Frustrated, I contacted the support team, but the responses were vague and unhelpful. No clear answers or solutions were offered, and my trust in the platform quickly started to erode. It became painfully clear that I wasn’t going to get anywhere with WHATS Invest, and I began to worry that my $10,000 might be lost for good. That's when I discovered Certified Recovery Services. Desperate to recover my funds, I decided to reach out to them for help. In just 24 hours, they worked tirelessly to recover the majority of my funds, successfully retrieving $8,500 85% of my initial investment. I couldn’t believe how quickly and efficiently they worked to get my money back. I’m extremely grateful for Certified Recovery Servicer's fast and professional service. Without them, I would have been left with a significant loss, and I would have had no idea how to move forward. If you find yourself in a similar situation with WHATS Invest or any other platform that isn’t delivering as promised, I highly recommend reaching out to Certified Recovery Services They were a lifesaver for me, helping me recover nearly all of my funds. It's reassuring to know that trustworthy services like this exist to help people when things go wrong. They also specialize in recovering money lost to online scams, so if you’ve fallen victim to such a scam, don’t hesitate to contact Certified Recovery Services they can help! Here's Their Info Below: WhatsApp: +1(740)258‑1417 mail: [email protected], [email protected] Website info; https://certifiedrecoveryservices.com
23.02.25 22:00 eunice49954

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 06:36 ANDREW DAVIS

RECOVER YOUR SCAMMED FUNDS AND CRYPTOCURRENCY VIA SPOTLIGHT RECOVERY Professional hackers at Spotlight Recovery provide services for compromised devices, accounts, and websites as well as for recovering stolen bitcoin and money from scams. They finish their work safely and quickly. Their order has been fulfilled since day one, and the victim will never be conscious of the outside entrance. Very few even attempt to give critical information, look into network security, or discreetly discuss personal issues. The Spotlight Recovery Crew helped me recover $264,000 that was stolen from my corporate bitcoin wallet, and I appreciate them giving me further details on the unidentified people. In the event that you have been defrauded of your hard-earned cash or bitcoins, contact SPOTLIGHT RECOVERY CREW at Contact: [email protected]
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
27.02.25 12:46 monikaguttmacher

Weddings are supposed to be magical, but the months leading up to mine were anything but. Already, wedding planning was a high-stress, sleep-deprived whirlwind: endless details to manage, from venue deposits and guest lists to dress fittings and vendor contracts. But nothing-and I mean, nothing-compared to the panic that washed over me when I realized that somehow, I had lost access to my Bitcoin wallet-with $600,000 inside. It happened in the worst possible way. In between juggling my to-do lists and trying to keep my sanity intact, I lost my seed phrase. I went through my apartment like a tornado, flipping through notebooks, checking every email, every file-nothing. I sat there in stunned silence, heart pounding, trying to process the fact that my entire savings, my security, and my financial future might have just vanished. In utter despair, I vented to my bridesmaid's group chat for some sympathetic words from the girls. Instead, one casually threw out a name that would change everything in a second: "Have you ever heard of Tech Cyber Force Recovery? They recovered Bitcoin for my cousin. You should call them." I had never heard of them before, but at that moment, I would have tried anything. I immediately looked them up, scoured reviews, and found story after story of people just like me—people who thought they had lost everything, only for Tech Cyber Force Recovery to pull off the impossible. That was all the convincing I needed. From the very first call, I knew I was in good hands. Their team was calm, professional, and incredibly knowledgeable. They explained the recovery process in a way that made sense, even through my stress-fogged brain. Every step of the way, they kept me informed, reassured me, and made me feel like this nightmare actually had a solution. And then, just a few days later, I got the message: "We have recovered your Bitcoin." (EMAIL. support @ tech cyber force recovery . com) OR WHATSAPP (+1 56 17 26 36 97) I could hardly believe my eyes: Six. Hundred. Thousand. Dollars. In my hands again. I let out my longest breath ever and almost cried, relieved. It felt like I woke up from a bad dream, but it was real, and Tech Cyber Force Recovery had done it. Because of them, I walked down the aisle not just as a bride, but as someone who had dodged financial catastrophe. Instead of spending my honeymoon stressing over lost funds, I got to actually enjoy it—knowing that my wallet, and my future, were secure. Would I refer to them? In a heartbeat. If you ever find yourself in that situation, please don't freak out, just call Tech Cyber Force Recovery. They really are the real deal.
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07.03.25 15:18 Ariduk

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08.03.25 04:42 andreassenhedda

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09.03.25 17:22 Wayne707

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10.03.25 18:18 springwilli

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10.03.25 20:19 wendytaylor015

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10.03.25 20:19 wendytaylor015

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11.03.25 16:23 dannywilliams

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11.03.25 16:23 dannywilliams

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05:35 cholevasaca

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